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BIBLIOGRAPHIC RECORD TARGET 

Graduate Library 
University of Michigan 

Preservation Office 

Storage Number: 



AAT1273 

ULFMTBRTaBLmT/C DT 07/15/88 R/DT 07/15/88 CC STATmmE/Ll 
010: : I a 05026791 
035/1: : | a (RLIN)MIUG84-B55766 
035/2: : | a (CaOTULAS)160189692 
040: : |aRPB |cRPB |dMiU 
050/1:0: |aQA9 |b.R93 
100:1 : I a Russell, Bertrand, | d 1872-1970. 

245:04: | a The principles of mathematics, | c by Bertrand Russell. | n vol. I. 
260: : | a Cambridge, | b University press, |cl903. 
300/1: : | a xxix, 534 p. | c 25 cm. 

500/1: : | a No more published. The material on the subject originally 
intended to form the 2d volume was later developed into an independent work: 
Principia mathematica, by A. N. Whitehead and Bertrand Russell, published in 3 
vols., Cambridge, 1910-13. 



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THE PRINCIPLES 

OF 

MATHEMATICS 



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Hotttiou : C. J. CLAY and SONS, 

CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, 

AVE MARIA LANE. 

ffilassoto: 50, WELLINGTON STREET. 



^ 


ii 


^ 


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1Letp>tg: F. A. BROCKHAUS. 

^zixs ^oxk: THE MACMILLAN COMPANY. 

Bomfeaa anti Calcutta: MACMILLAN AND CO., Ltd. 



[All Rights reserved.^ 



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THE PRINCIPLES 

OF 

MATHEMATICS 



BY 



BERTRAND RUSSELL M.A., 

LATE FELLOW OF TRINITY COLLEGE, CAMBRIDGE 



VOL I. 



Cambridge : 

at the University Press 

1903 



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^ambtitige: 

PRINTED BY J. AND C. F. CLAY, 
AT THE UNIVERSITY PRESS. 



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PREFACE. 

THE present work has two main objects. One of these, the proof 
that all pure mathematics deals exclusively with concepts definable 
in terms of a very small number of fundamental logical concepts, and 
that all its propositions are deducible from a very small number of 
fundamental logical principles, is undertaken in Parts II. — VII= of this 
Volume, and will be established by strict symbolic reasoning in Volume ii. 
The demonstration of this thesis has, if I am not mistaken, all the 
certainty and precision of which mathematical demonstrations are capable. 
As the thesis is very recent among mathematicians, and is almost 
universally denied by philosophers, I have undertaken, in this volume, 
to defend its various parts, as occasion arose, against such adverse 
theories as appeared most widely held or most difficult to disprove. 
I have also endeavoured to present, in language as untechnical as 
possible, the more important stages in the deductions by which the 
thesis is established. 

The other object of this work, which occupies Part I., is the 
explanation of the fundamental concepts which mathematics accepts 
as indefinable. This is a purely philosophical task, and I cannot flatter 
myself that I have done more than indicate a vast field of inquiry, and 
give a sample of the methods by which the inquiry may be conducted. 
The discussion of indefinables — which forms the chief part of philosophical 
logic — is the endeavour to see clearly, and to make others see clearly, 
the entities concerned, in order that the mind may have that kind of 
acquaintance with them which it has with redness or the taste of a 
pineapple. Where, as in the present case, the indefinables are obtained 
primarily as the necessary residue in a process of analysis, it is often 
easier to know that there must be such entities than actually to perceive 
them ; there is a process analogous to that which resulted in the discovery 
of Neptune, with the difference that the final stage — the search with a 
mental telescope for the entity which has been inferred — is often the 
most difficult part of the undertaking. In the case of classes, I must 
confess, I have failed to perceive any concept fulfilling the conditions 



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vi Preface 

requisite for the notion of class. And the contradiction discussed in 
Chapter x. proves that something is amiss, but what this is I have 
hitherto failed to discover. 

The second volume, in which I have had the great good fortune 
to secure the collaboration of Mr A. N. Whitehead, will be addressed 
exclusively to mathematicians ; it will contain chains of deductions, 
from the premisses of symbolic logic through Arithmetic, finite and 
infinite, to Geometry, in an order similar to that adopted in the present 
volume ; it will also contain various original developments, in which the 
method of Professor Peano, as supplemented by the Logic of Relations, 
has shown itself a powerful instrument of mathematical investigation. 

The present volume, which may be regarded either as a commentary 
upon, or as an introduction to, the second volume, is addressed in equal 
measure to the philosopher and to the mathematician ; but some parts 
will be more interesting to the one, others to the other. I should advise 
mathematicians, unless they are specially interested in Symbolic Logic, 
to begin with Part IV., and only refer to earlier parts as occasion arises. 
The following portions are more specially philosophical : Part I. 
(omitting Chapter ii.) ; Part II., Chapters xi., xv., xvi., xvii.; Part III.; 
Part IV., §207, Chapters xxvi,, xxvii., xxxi.; Part V., Chapters xli., 
xLii., xLiii.; Part VI., Chapters l., li., lii. ; Part VII., Chapters xiii., 
LI v., Lv., LVii., LViii. ; and the two Appendices, which belong to Part I., 
and should be read in connection with it. Professor Frege's work, which 
largely anticipates my own, was for the most part unknown to me when 
the printing of the present work began ; I had seen his Grimdgesetze 
der Arithmetik, but, owing to the great difficulty of his symbolism, I had 
failed to grasp its importance or to understand its contents. The only 
method, at so late a stage, of doing justice to his work, was to devote 
an Appendix to it ; and in some points the views contained in the 
Appendix differ from those in Chapter vi., especially in §§71, 73, 74. 
On questions discussed in these sections, I discovered errors after passing 
the sheets for the press; these errors, of which the chief are the denial 
of the null-class, and the identification of a term with the class whose 
only member it is, are rectified in the Appendices. The subjects 
treated are so difficult that I feel little confidence in my present 
opinions, and regard any conclusions which may be advocated as 
essentially hypotheses. 

A few words as to the origin of the present work may serve to 
show the importance of the questions discussed. About six years ago, 
I began an investigation into the philosophy of Dynamics. I was 
met by the difficulty that, when a particle is subject to several forces, 



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Preface vii 

no one of the component accelerations actually occurs, but only 
the resultant acceleration, of which they are not parts; this fact 
rendered illusory such causation of particulars by particulars as is 
affirmed, at first sight, by the law of gravitation. It appeared also that 
the difficulty in regard to absolute motion is insoluble on a relational 
theory of space. From these two questions I was led to a re-examination 
of the principles of Geometry, thence to the philosophy of continuity 
and infinity, and thence, with a view to discovering the meaning of the 
word any^ to Symbolic Logic. The final outcome, as regards the 
philosophy of Dynamics, is perhaps rather slender ; the reason of this 
is, that almost all the problems of Dynamics appear to me empirical, 
and therefore outside the scope of such a work as the present. Many 
very interesting questions have had to be omitted, especially in Parts 
yi. and VII., as not relevant to my purpose, which, for fear of 
misunderstandings, it may be well to explain at this stage. 

When actual objects are counted, or when Geometry and Dynamics 
are applied to actual space or actual matter, or when, in any other way, 
mathematical reasoning is applied to what exists, the reasoning employed 
has a form not dependent upon the objects to which it is applied being 
just those objects that they are, but only upon their having certain 
general properties. In pure mathematics, actual objects in the world 
of existence will never be in question, but only hypothetical objects 
having those general properties upon which depends whatever deduction 
is being considered ; and these general properties will always be 
expressible in terms of the fundamental concepts which I have called 
logical constants. Thus when space or motion is spoken of in pure 
mathematics, it is not actual space or actual motion, as we know them 
in experience, that are spoken of, but any entity possessing those abstract 
general properties of space or motion that are employed in the reasonings 
of geometry or dynamics. The question whether these properties belong, 
as a matter of fact, to actual space or actual motion, is irrelevant to pure 
mathematics, and therefore to the present work, being, in my opinion, 
a purely empirical question, to be investigated in the laboratory or the 
observatory. Indirectly, it is true, the discussions connected with pure 
mathematics have a very important bearing upon such empirical questions, 
since mathematical space and motion are held by many, perhaps most, 
philosophers to be self-contradictory, and therefore necessarily different 
from actual space and motion, whereas, if the views advocated in the 
following pages be valid, no such self-contradictions are to be found in 
mathematical space and motion. But extra-mathematical considerations 
of this kind have been almost wholly excluded from the present work. 



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viii Preface 

On fundamental questions of philosophy, my position, in all its chief 
features, is derived from Mr G. E, Moore. I have accepted from him 
the non-existential nature of propositions (except such as happen to 
assert existence) and their independence of any knowing mind; also 
the pluralism which regards the world, both that of existents and 
that of entities, as composed of an infinite number of mutually 
independent entities, with relations which are ultimate, and not 
reducible to adjectives of their terms or of the whole which these 
compose. Before learning these views from him, I found myself 
completely unable to construct any philosophy of arithmetic, whereas 
their acceptance brought about an immediate liberation from a large 
number of difficulties which I believe to be otherwise insuperable. 
The doctrines just mentioned are, in my opinion, quite indispensable 
to any even tolerably satisfactory philosophy of mathematics, as I hope 
the following pages will show. But I must leave it to my readers to 
judge how far the reasoning assumes these doctrines, and how far it 
supports them. Formally, my premisses are simply assumed; but the 
fact that they allow mathematics to be true, which most current 
philosophies do not, is surely a powerful argument in their favour. 

In Mathematics, my chief obligations, as is indeed evident, are to 
Georg Cantor and Professor Peano. If I had become acquainted 
sooner with the work of Professor Frege, I should have owed a 
great deal to him, but as it is I arrived independently at many 
results which he. had already established. At every stage of my work, 
I have been assisted more than I can express by the suggestions, the 
criticisms, and the generous encouragement of Mr A. N. Whitehead ; 
he also has kindly read my proofs, and greatly improved the final 
expression of a very large number of passages. Many useful hints 
I owe also to Mr W. E. Johnson ; and in the more philosophical parts 
of the book I owe much to Mr G. E. Moore besides the general position 
which underlies the whole. 

In the endeavour to cover so wide" a field, it has been impossible to 
acquire an exhaustive knowledge of the literature. There are doubtless 
many important works with which I am unacquainted ; but where the 
labour of thinking and writing necessarily absorbs so much time, such 
ignorance, however regrettable, seems not wholly avoidable. 

Many words will be found, in the course of discussion, to be defined 
in senses apparently departing widely from common usage. Such 
departures, I must ask the reader to believe, are never wanton, but have 
been made with great reluctance. In philosophical matters, they have 
been necessitated mainly by two causes. ♦First, it often happens that 



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Preface ix 

two cognate notions are both to be considered, and that language has 
two names for the one, but none for the other. It is then highly 
convenient to distinguish between the two names commonly used as 
synonyms, keeping one for the usual, the other for the hitherto nameless 
sense. The other cause arises from philosophical disagreement with 
received views. Where two qualities are commonly supposed inseparably 
conjoined, but are here regarded as separable, the name which has 
applied to their combination will usually have to be restricted to one 
or other. For example, propositions are commonly regarded as (1) true 
or false, (2) mental. Holding, as I do, that what is true or false is not 
in general mental, I require a name for the true or false as such, and 
this name can scarcely be other than proposition. In such a case, the 
departure from usage is in no degree arbitrary. As regards mathematical 
terms, the necessity for establishing the existence-theorem in each case — 
i.e. the proof that there are entities of the kind in question — has led to 
many definitions which appear widely different from the notions usually 
attached to the terms in question. Instances of this are the definitions 
of cardinal, ordinal and complex numbers. In the two former of these, 
and in many other cases, the definition as a class, derived from the 
principle of abstraction, is mainly recommended by the fact that it 
leaves no doubt as to the existence -theorem. But in many instances of 
such apparent departure from usage, it may be doubted whether more 
has been done than to give precision to a notion which had hitherto 
been more or less vague. 

For publishing a work containing so many unsolved difficulties, my 
apology is, that investigation revealed no near prospect of adequately 
resolving the contradiction discussed in Chapter x., or of acquiring a 
better insight into the nature of classes. The repeated discovery of errors 
in solutions which for a time had satisfied me caused these problems to 
appear such as would have been only concealed by any seemingly satis- 
factory theories which a slightly longer reflection might have produced ; 
it seemed better, therefore, merely to state the difficulties, than to wait 
until I had become persuaded of the truth of some almost certainly 
erroneous doctrine. 

My thanks are due to the Syndics of the University Press, and to 
their Secretary, Mr R. T. Wright, for their kindness and courtesy 
in regard to the present volume. 



L0ND0N_, 

December, 1902. 



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TABLE OF CONTENTS 



Preface 



PAGE 
V 



PART I. 
THE INDEFINABLES OE MATHEMATICS. 



CHAPTER I. 
DEFINITION OF PURE MATHEMATICS. 

1. Definition of pure mathematics ........ 3 

2. The principles of mathematics are no longer controversial . . . 3 

3. Pure mathematics uses only a few notions^, and these are logical 

constants ........... 4 

4. All pure mathematics follows formally from twenty premisses . . 4 

5. Asserts formal implications ......... 5 

6. And employs variables ......... 5 

7. Which may have any value without exception ..... 6 

8. Mathematics deals with types of relations . . . . . . 7 

9. Applied mathematics is defined by the occurrence of constants which 

are not logical . . . . . . . . . .8 

10. Relation of mathematics to logic 8 



CHAPTER II. 

SYMBOLIC LOGIC. 

11. Definition and scope of symbolic logic 10 

12. The indefinables of symbolic logic 10 

13. Symbolic logic consists of three parts . . . . . . . 11 

A. The Propositional Calculus. 

14. Definitioii ,.^ 13 

15. Distinctitm 'hetv^'een implication and formal implication ... 14 

16. Implica'^i®ii jj^Sefinable 14 

17. Two indi^ilfebles and ten primitive propositions in this calculus . . 15 

18. The teii^pll^itiye propositions ........ 16 

19. Disjun^5^'-'ltnd negation defined 17 



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20. 
21. 
22. 
23. 
24. 
25. 
26. 



27. 
28. 
29. 
30. 



31. 
32. 
33. 
34. 
35. 
36. 



B. The Calculus of Classes. 

PAGE 

Three new indefinables . . . . . . . . . 18 

The relation of an individual to its class . . . . . .19 

Propositional functions ......... 19 

The notion of such that ......... 20 

Two new primitive propositions ........ 20 

Relation to propositional calculus ....... 21 

Identity 23 

C. The Calculus of Relations. 

The logic of relations essential to mathematics ..... 23 

New primitive propositions ......... 24 

Relative products .......... 25 

Relations with assigned domains ........ 26 

D. Peands Symbolic Logic. 

Mathematical and philosophical definitions ...... 26 

Peano's indefinables . . . . . . . . . . 27 

Elementary definitions . . . . . . . . .28 

Peano's primitive propositions ........ 29 

Negation and disjunction ......... 31 

Existence and the null-class ........ 32 



CHAPTER III. 
IMPLICATION AND FORMAL IMPLICATION. 

37. Meaning of implication ......... 33 

38. Asserted and unasserted propositions ....... 34 

39. Inference does not require two premisses . . . . . . , 35 

40. Formal implication is to be interpreted extensionally .... 36 

41. The variable in a formal implication has an unrestricted field . . 36 

42. A formal implication is a single propositional function^ not a relation 

of two 38 

43. Assertions ............ 39 

44. Conditions that a term in an implication may be varied . • . . 39 

45. Formal implication involved in rules of inference .... 40 



CHAPTER IV. 



PROPER NAMES, ADJECTIVES AND VERBS. 

46. Proper names, adjectives and verbs distinguished 

47. Terms ........ 

48. Things and concepts . . . 

49. Concepts as such and as terms 

50. Conceptual diversity ..... 

51. Meaning and the subject-predicate logic 

52. Verbs and truth . 

53. Air verbs, except perhaps is, express relations 

54. Relations joer se and relating relations 
65. Relations are not particularized by their terms 



42 
43 
44 
45 
46 
47 
47 
49 
49 
50 



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CHAPTER V. 
DENOTING. 

PAGE 

^^. Definition of denoting .......... 53 

57. Connection with subject-predicate propositions ..... 54 

58. Denoting concepts obtained from predicates . .... 5^ 

59. Extensional account of all, every, any, a and some . . . . 6Q 

60. Intensional account of the same ........ 58 

61. Illustrations 59 

62. The difference between all, every, etc. lies in the objects denoted^ not 

in the way of denoting them . . . . . . . 61 

63. The notion of the and definition ........ 62 

64. ITie notion of the and identity ........ 63 

Q6. Summary ............ 64 



CHAPTER VI. 



CLASSES. 

QQ. Combination of intensional and extensional standpohits required 

67. Meaning of class ..... 

68. Intensional and extensional genesis of classes 

69. Distinctions overlooked by Peano 

70. The class as one and as many 

71. The notion of and ..... 

72. All men is not analyzable into all and men . 

73. There are null class-concepts^ but there is no null class 

74. The class as one^ except when it has one term^ is distinct from the 

class as many . . . 

75. Every, any, a and sorne each denote one object^ but an ambiguous one 

76. The relation of a term to its class ...... 

77. The relation of inclusion between classes ..... 

78. The contradiction . . . . . . . 

79. Summary ........... 



66 
67 
67 
68 
68 
69 
72 
73 

76 

77 
77 
78 
79 
80 



CHAPTER VII. 
PROPOSITIONAL FUNCTIONS. 

80. Indefinability of such that 82 

81. Where a fix^d relation to a fixed term is asserted^ a propositional 

funo;tion can "be analyzed into a variable subject and a constant 

assertic^v- ........... 83 

82. But this analysis is impossible in other cases ..... 84 

83. Variation of the ^:oncept in a proposition 86 

84. Relation of propoattional functions to classes . . . . . 88 

85. A propositional fij.^ction is in general not analyzable into a constant 

and a variable element . . . . . . . . 88 



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CHAPTER VIII. 



THE VARIABLE. 



86. Nature of the variable 

87. Relation of the variable to any . 

88. Formal and restricted variables . 

89. Formal implication presupposes any 

90. Duality of any and so7ne 

91. The clsiss-concei^i p7'opositional function is indefinable 

92. Other classes can be defined by means of such that 

93. Analysis of the variable ..... 



PAGE 

89 
89 
91 
91 
92 
92 
93 
93 



CHAPTER IX. 

RELATIONS. 

94. Characteristics of relations ....... 

95. Relations of terms to themselves . . . . . 

96. The domain and the converse domain of a relation 

97. Logical sum^ logical product and relative product of relations 

98. A relation is not a class of couples ..... 

99. Relations of a relation to its terms ..... 



95 
96 
97 
98 
99 
99 



CHAPTER X. 
THE CONTRADICTION. 

100. Consequences of the contradiction . . . . . . . 101 

101. Various statements of the contradiction . . . . . . 102 

102. An analogous generalized argument ....... 102 

103. Variable propositional functions are in general inadmissible . . 103 

104. The contradiction arises from treating as one a class which is only 

many ............ 104 

105. Other jon7?m/«6*fe possible solutions appear inadequate . . . .105 

106. Summary of Part I 106 



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Table of Contents 

PAET 11. 
NUMBER. 



CHAPTER XI. 
DEFINITION OF CARDINAL NUMBERS. 

PAGE 

107. Plan of Part II ... . . Ill 

108. Mathematical meaning of definition . . . , . . .111 

109. Definition of numbers by abstraction . . . . . . . 112 

110. Objections to this definition . . . . . . . . 114 

111. Nominal definition of numbers ........ 115 



CHAPTER XII. 
ADDITION AND MULTIPLICATION. 



112. Only integers to be considered at present 

113. Definition of arithmetical addition . . . 

114. Dependence upon the logical addition of classes . 

115. Definition of multiplication ..... 

116. Connection of addition, multiplication and exponentiation 



iir 

117 

118- 
119' 
119 



CHAPTER XIII. 
FINITE AND INFINITE. 



117. Definition of finite and infinite . ... 

118. Definition of ao . . . . . . , . . 

119. Definition of finite numbers by mathematical induction 



121 
121 
123 



CHAPTER XIV. 
THEORY OF FINITE NUMBERS. 

120. Peano's indefinables and primitive propositions \ . 

121. Mutual independence of t^e latter . . . 

122. Peano really defines progressions^ not finite numbers . 

123. Proof of Peano's primitiye propositions 



124 
125 
125 
127 



R. 



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CHAPTER XV. 



ADDITION OF TERMS AND ADDITION OF CLASSES. 

124. Philosophy and mathematics distinguivshed ..... 

125. Is there a more fundamental sense of number than that defined above 

126. Numbers must be classes . . . . 

127. Numbers apply to classes as many .... 

128. One is to be asserted, not of terms, but of unit classes 

129. Counting not fundamental in arithmetic , . 

130. Numerical conjunction and plurality .... 

131. Addition of terms generates classes primarily, not numbers 

132. A term is indefinable, but not the number 1 



PAGE 

129 
130 
131 
132 
132 
133 
133 
135 
135 



CHAPTER XVI. 



WHOLE AND PART. 

133. Single terms may be either simple or complex 

134. Whole and part cannot be defined by logical priority . 

135. Three kinds of relation of whole and part distinguished 

136. Two kinds of wholes distinguished ..... 

137. A whole is distinct from the numerical conjunction of its parts 

138. How far analysis is falsification ...... 

139. A class as one is an aggregate ...... 



137 
137 
138 
140 
141 
141 
141 



CHAPTER XVII. 
INFINITE WHOLES. 

140. Infinite aggregates must be admitted . 

141. Infinite unities, if there are any, are unknown to us 

142. Are all infinite wholes aggregates of terms ? 

143. Grounds in favour of this view .... 



143 
144 
146 
146 



CHAPTER XVIII. 

RATIOS AND FRACTIONS. 

144. Definition of ratio . . . . . . . . . .149 

145. Ratios are one-one relations . . . . . . . .150 

146. Fractions are concerned with relations of whole and part . . .150 

147. Fractions depend, not upon number, but upon magnitude of divisibility 151 

148. Summary of Part II . . . . 152 



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xvii 



PART III. 
QUANTITY. 



CHAPTER XIX. 

THE MEANING OF MAGNITUDE. 

149. Previous views on the relation of number and quantity 

150. Quantity not fundamental in mathematics . 

151. Meaning- of magnitude and quantity 

152. Three possible theories of equality to be examined 

153. Equality is not identity of number of parts 

154. Equality is not an unanalyzable relation of quantities 

155. Equality is sameness of magnitude 

156. Every particular magnitude is simple 

157. The principle of abstraction 

158. Summary ..... 
Note 



PAGE 
157 

158 
159 
159 
160 
162 
164 
164 
166 
167 
168 



CHAFTEK XX. 
THE RANGE OF QUANTITY. 

159. Divisibility does not belong to all quantities ..... 170 

160. Distance l7l 

161. Differential coefficients . . . . . . . . .173 

162. A magnitude is never divisible^ but may be a magnitude of divisibility . 173 

163. Every magnitude is unanalyzable . . . . . . .174 

CHAPTEK XXI. 

NUMBERS AS EXPRESSING MAGNITUDES : MEASUREMENT. 

164. Definition of measurement . . . . . . . ... 176 

165. Possible grounds for holding all magnitudes to be measurable . . 176 

166. Intrinsic measurability ......... 177 

167. Of divisibilities .178 

168. And of distances .179 

169. Measure of distance and measure of stretch ...... 181 

170. Distance-theories and stretch-theories of geometry . . . .181 

171. Extensive and intensive magnitudes . . . . . . .182 





CHAPTER XXII. 






ZERO. 




172. 


Difficulties as to zero ......... 


184 


173. 


Meinong's theory 


184 


174. 


Zero as minimum 


185 


175. 


Zero distance as identity ........ 


186 


176. 


Zero as a null segment ........ 


186 


177. 


Zero and negation 


186 


178. 


Every kind of zero magnitude is in a sense indefinable . 


187 



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CHAPTER XXIII. 
INFINITY, THE INFINITESIMAL, AND CONTINUITY. 



179. Problems of infinity not specially quantitative 

180. Statement of the problem in regard to quantity . 

181. Three antinomies ....... 

182. Of which the antitheses depend upon an axiom of finitude 

183. And the use of mathematical induction 

184. Which are both to be rejected 

185. Provisional sense of continuity ..... 

186. Summary of Part III 



PAGE 
188 

188 
189 
190 
192 
192 
193 
194 



PART IV. 
ORDER. 



CHAPTER XXIV. 
THE GENESIS OF SERIES. 



187. Importance of order .... 

188. Between and separation of couples 

189. Generation of order by one-one relations 

190. ^Y transitive asymmetrical relations . 

191. By distances ..... 

192. By triangular relations 

193. By relations between asymmetrical relations 

194. And by separation of couples 



199 
199 
200 
203 
204 
204 
205 
205. 



CHAPTER XXV. 

THE MEANING OF ORDER. 



195. What is order } , 

196. Three theories of hetween 

197. First theory .... 

198. A relation is not hetween its terms 

199. Second theory of hetween 

200. There appear to be ultimate triangular relations 

201. Reasons for rejecting the second theory 

202. Third theory of hetween to be rejected 

203. Meaning of separation of couples 

204. Reduction to transitive asymmetrical relations 

205. This reduction is formal .... 

206. But is the reason why separation leads to order 

207. The second way of generating series is alone fundamental,, and gives 

the meaning of order ......... 



207 
207 
208 
210 
211 
211 
213 
213 
214 
215 
216 
216 

216 



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CHAPTER XXVI. 



ASYMMETRICAL RELATIONS. 

208. Classification of relations as regards symmetry and transitiveness 

209. Symmetrical transitive relations . 

210. Reflexiveness and the principle of abstraction 

211. Relative position ..... 

212. Are relations reducible to predications } 

213. Monadistic theory of relations 

214. Reasons for rejecting this theory 

215. Monistic theory and the reasons for rejecting it 

216. Order requires that relations should be ultimate 



PAGE 

218 
219 
219 
220 
221 
• 222 
222 
224 
226 



CHAPTER XXVII. 

DIFFERENCE OF SENSE AND DIFFERENCE OF SIGN. 

217. Kant on difference of sense ........ 227 

218. Meaning of difference of sense . . . . . . . . 228 

219. Difference of sign 228 

220. In the cases of finite numbers . . . . . . . . 229 

221. And of magnitudes 229 

222. Right and left . 231 

223. Difference of sign arises from difference of sense among transitive 

asymmetrical relations ........ 232 



CHAPTER XXVIII. 

ON THE DIFFERENCE BETWEEN OPEN AND CLOSED SERIES. 

224. What is the difference between open and closed series ? . . . 234 

225. Finite closed series .......... 234 

226. Series generated by triangular relations ...... 236 

227. Four-term relations 237 

228. Closed series are such as have an arbitrary first term .... 238 



229. 
230, 
231. 
232. 
233. 



CHAPTER XXIX. 

PROGRESSIONS AND ORDINAL NUMBERS. 

D ^finition of progressions . . ....... 239 

All finite arithmetic applies to every progression .... 240 

Definition of ordinal numbers . . . . . . . . 242 

Definition of '^ nth" 243 

Positive and negative ordinals ........ 244 



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CHAPTER XXX. 
DEDEKIND'S THEORY OF NUMBER. 



234. Dedekind's principal ideas . 

235. Representation of a system . 

236. The notion of a chain 

237. The chain of an element 

238. Generalized form of mathematical induction 

239. Definition of a singly infinite system . 

240. Definition of cardinals 

241. Dedekind's proof of mathematical induction 

242. Objections to his definition of ordinals 

243. And of cardinals .... 



PAGE 

245 
245 
246 
246 
246 
247 
247 
248 
248 
249 



CHAPTER XXXI. 

DISTANCE. 

244. Distance not essential to order ........ 252 

245. Definition of distance .......... 253 

246. Measurement of distances ......... 254 

247. In most series^ the existence of distances is doubtful .... 254 

248. Summary of Part IV 2.55 



PART V. 
INFINITY AND CONTINUITY. 



CHAPTER XXXII. 
THE CORRELATION OF SERIES. 

249. The infinitesimal and space are no longer required in a statement of 

principles ........ 

250. The supposed contradictions of infinity have been resolved 

251. Correlation of series ...... 

252. Independent series and series by correlation 

253. Likeness of relations . . . . . . 

254. Functions ........ 

255. Functions of a variable whose values form a series 

256. Functions which are defined by formulae 

257. Complete series . . . . . . 



259 
260 
260 
262 

262 
263 
264 

267 
269 



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CHAPTER XXXIII. 
REAL NUMBERS. 

PAGE 

258. Real numbers are not limits of series of rationals .... 270 

259. Segments of rationals ......... 271 

260. Properties of segments ......... 272 

261. Coherent classes in a series . . . . . . . . 274 

Note . 274 



CHAPTER XXXIV. 



LIMITS AND IRRATIONAL NUMBERS. 



262. Definition of a limit ...... 

263. Elementary properties of limits .... 

264. An arithmetical theory of irrationals is indispensable 

265. Dedekind's theory of irrationals . 

266. Defects in Dedekind's axiom of continuity 

267. Objections to his theory of irrationals 

268. Weierstrass's theory .... 

269. Cantor's theory ..... 

270. Real numbers are segments of rationals 



276 
277 
277 
278 
279 
280 
282 
283 
285 



CHAPTER XXXV. 

CANTOR'S FIRST DEFINITION OF CONTINUITY. 

271. The arithmetical theory of continuity is due to Cantor . . . 287 

272. Cohesion .288 

273. Perfection 290 

274. Defect in Cantor's definition of perfection ...... 291 

275. The existence of limits must not be assumed without special grounds . 293 



CHAPTER XXXVI. 

ORDINAL CONTINUITY. 

276. Continuity is a purely ordinal notion ...... 296 

277. Cantor's ordinal definition of continuity . . . . . . 296 

278. Only ordinal notions occur in this definition ..... 298 

279. Infinite classes of integers can be arranged in a continuous series . 298 

280. Segments of general compact series ....... 299 

281. Segments defined by fundamental series ...... 300 

282. Two compact series may be combined to form a series which is not 

compact ........... 303 



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CHAPTER XXXVII. 
TRANSFINITE CARDINALS. 

PAGE 

283. Transfinite cardinals diifer widely from traiisfinite ordinals . . . 304 

284. Definition of cardinals . . . . . . . • • 304 

285. Properties of cardinals ......... 306 

286. Addition^ multiplication and exponentiation 307 

287. The smallest transfinite cardinal a^ 309 

288. Other transfinite cardinals . . . . . . . . . 310 

289. Finite and transfinite cardinals form a single series by relation to 

greater and less ......... 311 



CHAPTER XXXVIII. 
TRANSFINITE ORDINALS. 

290. Ordinals are classes of serial relations . . . . . .312 

291. Cantor's definition of the second class of ordinals .... 312 

292. Definition of o) 314 

293. An infinite class can be arranged in many types of series . . .315 

294. Addition and subtraction of ordinals . . . . . • . 317 

295. Multiplication and division , . . . . . . . .318 

296. Well-ordered series 319 

297. Series which are not well-ordered ....... 320 

298. Ordinal numbers are types of well-ordered series .... 321 

299. Relation-arithmetic 321 

300. Proofs of existence-theorems . . . . . . . . 322 

301. There is no maximum ordinal number . . . . . . 323 

302. Successive derivatives of a series . . . . . . . . 323 



CHAPTER XXXIX. 
THE INFINITESIMAL CALCULUS. 

303. The infinitesimal has been usually supposed essential to the calculus . 325 

304. Definition of a continuous function ....... 326 

305. Definition of the derivative of a function . ... . . . 328 

306. The infinitesimal is not implied in this definition .... 329 

307. Definition of the definite integral . . . . . . . 329 

308. Neither the infinite nor the infinitesimal is involved in this definition . 330 



CHAPTER XL. 
THE INFINITESIMAL AND THE IMPROPER INFINITE. 



309. A precise definition of the infinitesimal is seldom given 

310. Definition of the infinitesimal and the improper infinite 

311. Instances of the infinitesimal ..... 

312. No infinitesimal segments in compact series 

313. Orders of infinity and infinitesimality .... 

314. Summary ......... 



331 
331 
332 
334 
335 
337 



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CHAPTER XLI, 
PHILOSOPHICAL ARGUMENTS CONCERNING THE INFINITESIMAL. 

PAGE 

315. Current philosophical opinions illustrated by Cohen .... 338 

316. Who bases the calculus upon infinitesimals ...... 338 

317. Space and motion are here irrelevant . . . . . . . 339 

318. Cohen regards the doctrine of limits as insufficient for the calculus . 339 

319. And supposes limits to be essentially quantitative .... 340 

320. To involve infinitesimal differences ....... 341 

321. And to introduce a new meaning of equality . . . . . 341 

322. He identifies the inextensive with the intensive . . . . ► . 342 

323. Consecutive numbers are supposed to be required for continuous 

change . 344 

324. Cohen's views are to be rejected . . . . . . . . 344 



CHAPTER XLII. 



THE PHILOSOPHY OF THE CONTINUUM. 

325. Philosophical sense of continuity not here in question 

326. The continuum is composed of mutually external units 

327. Zeno and Weierstrass ...... 

328. The argument of dichotomy 

329. The objectionable and the innocent kind of endless regress 

330. Extensional and intensional definition of a whole 

331. Achilles and the tortoise .... 

332. The arrow 

333. Change does not involve a state of change . 

334. The argument of the measure 

335. Summary of Cantor's doctrine of continuity 

336. The continuum consists of elements 



346 
346 
347 
348 
348 
349 
350 
350 
351 
352 
353 
353 



CHAPTER XLIIL 

THE PHILOSOPHY OF THE INFINITE. 

337. Historical retrospect .......... 355 

338. Positive doctrine of the infinite ........ 356 

339. Proof that there are infinite classes ....... 357 

340. The paradox of Tristram Shandy ....... 358 

341. A whole and a part may be similar . . . . . . . 359 

342. Whole and part and formal implication . . . . . . . 360 

343. No immediate predecessor of oo or a^ . . . . . . . 361 

344. Diflficulty as regards the number of all terms^ objects^ or propositions . 362 

345. Cantor's first proof that there is no greatest number .... 363 

346. His second proof 364 

347. Every class has more sub-classes than terms . . . . . 366 

348. But this is impossible in certain cases ....... 366 

349. Resulting contradictions ......... 367 

350. Summary of Part V 368 



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PART VI. 
SPACE. 



CHAPTER XLIV. 
DIMENSIONS AND COMPLEX NUMBERS. 

PAGE 

351. Retrospect 371 

352. Geometry is the science of series of two or more dimensions . . 372 

353. Non-Euclidean geometry 372 

354. Definition of dimensions ......... 374 

355. Remarks on the definition . ........ 375 

356. The definition of dimensions is purely logical ..... 376 

357. Complex numbers and universal algebra ...... 376 

358. Algebraical generalization of number ....... 377 

359. Definition of complex numbers ........ 378 

360. Remarks on the definition 379 



CHAPTER XLV. 

PROJECTIVE GEOMETRY. 

361. Recent threefold scrutiny of geometrical principles . . . . 381 

362. Projective^ descriptive and metrical geometry ..... 381 

363. Projective points and straight lines . . . . . . . 382 

364. Definition of the plane 384 

365. Harmonic ranges .......... 384 

366. Involutions 385 

367. Projective generation of order ........ 386 

368. M5bius nets . .388 

369. Projective order presupposed in assigning irrational coordinates . 389 

370. Anharmonic ratio .......... 390 

371. Assignment of coordinates to any point in space .... 390 

372. Comparison of projective and Euclidean geometry .... 391 

373. The principle of duality 392 



CHAPTER XLVI. 

DESCRIPTIVE GEOMETRY. 

374. Distinction between projective and descriptive geometry 

375. Method of Pasch and Peano .... 

376. Method employing serial relations 

377. Mutual independence of axioms .... 

378. Logical definition of the class of descriptive spaces 



393 
394 
395 
396 
397 



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PAGE 

379. Parts of straight lines 397 

380. Definition of the plane .398 

381. Solid geometry 399 

382. Descriptive geometry applies to Euclidean and hyperbolic^ but not 

elliptic space .......... 399 

383. Ideal elements 400 

384. Ideal points 400 

885. Ideal lines 401 

386. Ideal planes 402 

387. The removal of a suitable selection of points renders a projective space 

descriptive ........... 403 



CHAPTER XLYII. 
METRICAL GEOMETRY. 

388. Metrical geometry presupposes projective or descriptive geometry . 404 

389. Errors in Euclid 404 

390. Superposition is not a valid method ....... 405 

391. Errors in Euclid (continued) 406 

392. Axioms of distance 407 

393. Stretches 408 

394. Order as resulting from distance alone ...... 409 

395. Geometries which derive the straight line from distance . . . 410 

396. In most spaces^ magnitude of divisibility can be used instead of 

distance . . . . . . . . . • • 411 

397. Meaning of magnitude of divisibility . . . . . . .411 

398. DiflSculty of making distance independent of stretch .... 413 

399. Theoretical meaning of measurement . . . . . . .414 

400. Definition of angle 414 

401. Axioms concerning angles . . . . . . . . .415 

402. An angle is a stretch of rays^ not a class of points .... 416 

403. Areas and volumes . . . . . . . . . . 417 

404. Right and left 417 



CHAPTER XLVIII. 



RELATION OF METRICAL TO PROJECTIVE AND 
DESCRIPTIVE GEOMETRY. 

405. Non-quantitative geometry has no metrical presuppositions 

406. Historical development of non-quantitative geometry 

407. Non-quantitative theory of distance 

408. In descriptive geometry .... 

409. And in projective geometry 

410. Geometrical theory of imaginary point-pairs 

411. New projective theory of distance 



419 
420 
421 
423 
425 
426 
427 



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CHAPTER XLIX. 
DEFINITIONS OF VARIOUS SPACES. 

412. All kinds of spaces are definable in purely logical terms 

413. Definition of projective spaces of three dimensions 

414. Definition of Euclidean spaces of three dimensions 

415. Definition of CliiFord's spaces of two dimensions . 



PAGE 

429 
480 
432 
434 



CHAPTER L. 

THE CONTINUITY OF SPACE. 

416. The continuity of a projective space ....... 437 

417. The continuity of a metrical space ....... 438 

418. An axiom of continuity enables us to dispense with the postulate of the 

circle . 440 

419. Is space prior to points .^ ........ . 440 

420. Empirical premisses and induction ....... 441 

421. There is no reason to desire our premisses to be self-evident . . 441 

422. Space is an aggregate of points^ not a unity ..... 442 



CHAPTER LI. 

LOGICAL ARGUMENTS AGAINST POINTS. 

423. Absolute and relative position 

424. Lotze's arguments against absolute position . 

425. Lotze's theory of relations . . 

426. The subject-predicate theory of propositions 

427. Lotze's three kinds of Being 

428. Argument from the identity of indiscernibles 

429. Points are not active ..... 

430. Argument from the necessary truths of geometry 

431. Points do not imply one another . 



445 
446 
446 
448 
449 
451 
452 
454 
454 



CHAPTER LII. 
KANT'S THEORY OF SPACE. 

432. The present work is diametrically opposed to Kant .... 456 

433. Summary of Kant's theory ......... 456 

434. Mathematical reasoning requires no extra-logical element . . . 457 

435. Kant's mathematical antinomies ........ 458 

436. Summary of Part VI 461 



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PART YIL 
MATTER AND MOTION. 



CHAPTER LIII. 
MATTER. 

PAGE 

437. Dynamics is here considered as a branch of pure mathematics . . 465 

438. Matter is not implied by space ........ 465 

439. Matter as substance 466 

440. Relations of matter to space and time ....... 467 

441. Definition of matter in terms of logical constants .... 468 



CHAPTER LIV. 

MOTION. 

442. Definition of change .......... 469 

443. There is no such thing as a state of change ...... 471 

444. Change involves existence ......... 471 

445. Occupation of a place at a time ........ 472 

446. Definition of motion . . . . . . . . . . 472 

447. There is no state of motion ......... 473 



CHAPTER LV. 
CAUSALITY. 



448. The descriptive theory of dynamics 

449. Causation of particulars by particulars 

450. Cause and effect are not temporally contiguous . 

451. Is there any causation of particulars by particulars .^^ 

452. Generalized form of causality . . 



474 
475 
476 
477 
478 



CHAPTER LVI. 
DEFINITION OF A DYNAMICAL WORLD. 



453. Kinematical motions 

454. Kinetic motions . 



480 
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CHAPTER LVII. 



NEWTON'S LAWS OF MOTION. 



455. Force and acceleration are fictions 

456. The law of inertia 

457. The second law of motion . 

458. The third law ... . 

459. Summary of Newtonian principles 

460. Causality in dynamics 

461. Accelerations as caused by particulars 

462. No part of the laws of motion is an a priori truth 



PAGE 

482 
482 
483 
483 
485 
486 
487 
488 



CHAPTER LVin. 
ABSOLUTE AND RELATIVE MOTION. 

463. Newton and his critics 489 

464. Grounds for absolute motion 490 

465. Neumann's theory .......... 490 

466. Streintz's theory 491 

467. Mr Macaulay's theory 491 

468. Absolute rotation is still a change of relation ..... 492 

469. Mach's reply to Newton 492 



CHAPTER LIX. 
HERTZ'S DYNAMICS. 

470. Summary of Hertz's system 494 

471. Hertz's innovations are not fundamental from the point of view of 

pure mathematics ......... 495 

472. Principles common to Hertz and Newton . . . . . . 496 

473. Principle of the equality of cause and effect , . . . . . 496 

474. Summary of the work ......... . 497 



APPENDIX A. 
THE LOGICAL AND ARITHIVIETICAL DOCTRINES OF FREGE. 

475. Principal points in Frege's doctrines . . . . . . . 501 

476. Meaning- and indication . . . . . . . . . 502 

477. Truth-values and judgment ........ 502 

478. Criticism 503 

479. Are assumptions proper names for the true or the false } . . , 504 

480. Functions 505 

481. Begriff and Gegenstand ......... 507 



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PAGE 

482. Recapitulation of theory of propositioiial functions .... 508 

483. Can concepts be made logical subjects ?...... 510 

484. Ranges . . . .510 

485. Definition of e and of relation ........ 512 

486. Reasons for an extensional view of classes ...... 513 

487. A class which has only one member is distinct from its only member . 513 

488. Possible theories to account for this fact ...... 514 

489. Recapitulation of theories already discussed . . . . .515 

490. The subject of a proposition may be plural ...... 516 

491. Classes having only one member . . . . . . . .517 

492. Theory of types 518 

493. Implication and symbolic logic . . . . . . . .518 

494. Definition of cardinal numbers ........ 519 

495. Frege's theory of series . . . . . . . . . 520 

496. Kerry's criticisms of Frege ......... 520 



APPENDIX B. 

THE DOCTRINE OF TYPES. 

497. Statement of the doctrine ......... 523 

498. Numbers and propositions as types . . . . . . . 525 

499. Are propositional concepts individuals .^ ..... . 526 

500. Contradiction arising from the question whether there are more classes 

of propositions than propositions . . . . . . . 527 

Index .............. 529 



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PART I. 

THE INDEFINABLES OF MATHEMATICS. 



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CHAPTER I. 



DEFINITION OF PURE MATHEMATICS. 

1. Pure Mathematics is the class of all propositions of the form 
'' p implies g,'' where p and q are propositions containing one or more 
variables, the same in the two propositions, and neither p nor q contains 
any constants except logical constants. And logical constants are all 
notions definable in terms of the following : Implication, the relation 
of a term to a class of which it is a member, the notion of mich that, 
the notion of relation, and such further notions as may be involved 
in the general notion of propositions of the above form. In addition 
to these, mathematics uses a notion which is not a constituent of the 
propositions which it considers, namely the notion of truth. 

2. The above definition of pure mathematics is, no doubt, some- 
what unusual. Its various parts, nevertheless, appear to be capable of 
exact justification — -a justification which it will be the object of the 
present work to provide. It will be shown that whatever has, in the 
past, been regarded as pure mathematics, is included in our definition, 
and that whatever else is included possesses those marks , by which 
mathematics is commonly though vaguely distinguished from other 
studies. The definition professes to be, not an arbitrary decision to 
use a common word in an uncommon signification, but rather a precise 
analysis of the ideas which, more or less unconsciously, are implied in 
the ordinary employment of the term. Our method will therefore be 
one of analysis, and our problem may be called philosophical — in the 
sense, that is to say, that we seek to pass from the complex to the 
simple, from the demonstrable to its indemonstrable premisses. But 
in one respect not a few of our discussions will differ from those that 
are usually called philosophical. We shall be able, thanks to the labours 
of the mathematicians themselves, to arrive at certainty in regard to 
most of the questions with which we shall be concerned ; and among 
those capable of an exact solution we shall find many of the problems 
which, in the past, have been involved in all the traditional uncertainty 
of philosophical strife. The nature of number, of infinity, of space, 
time and motion, and of mathematical inference itself, are all questions 

1—2 



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4 The Indefinables of Mathematics [chap, i 

to which, in the present work, an answer professing itself demonstrable 
with mathematical certainty will be given — an answer which, however, 
consists in reducing the above problems to problems in pure logic, 
which last will not be found satisfactorily solved in what follows. 

3. The Philosophy of Mathematics has been hitherto as con- 
troversial, obscure and unprogressive as the other branches of philosophy. 
Although it was generally agreed that mathematics is in some sense 
true, philosophers disputed as to what mathematical propositions really 
meant : although something was true, no two people were agreed as to 
what it was that was true, and if something was known, no one knew^ 
what it was that was known. So long, however, as this was doubtful, 
it could hardly be said that any certain and exact knowledge was to be 
obtained in mathematics. We find, accordingly, that idealists have 
tended more and more to regard all mathematics as dealing with mere 
appearance, while empiricists have held everything mathematical to be 
approximation to some exact truth about which they had nothing to 
tell us. This state of things, it must be confessed, was thoroughly 
unsatisfactory. Philosophy asks of Mathematics : What does it mean ? 
Mathematics in the past was unable to answer, and Philosophy answered 
by introducing the totally irrelevant notion of mind. But now 
Mathematics is able to answer, so far at least as to reduce the whole 
of its propositions to certain fundamental notions of logic. At this 
point, the discussion must be resumed by Philosophy. I shall endeavour 
to indicate what are the fundamental notions involved, to prove at 
length that no others occur in mathematics, and to point out briefly 
the philosophical difficulties involved in the analysis of these notions. 
A complete treatment of these difficulties would involve a treatise on 
Logic, which will not be found in the following pages. 

4. There was, until very lately, a special difficulty in the principles 
of mathematics. It seemed plain that mathematics consists of deductions, 
and yet the orthodox accounts of deduction were largely or wholly 
inapplicable to existing mathematics. Not only the Aristotelian 
syllogistic theory, but also the modern doctrines of Symbolic Logic, 
were either theoretically inadequate to mathematical reasoning, or at 
any rate required such artificial forms of statement that they could not 
be practically applied. In this fact lay the strength of the Kantian 
view, which asserted that mathematical reasoning is not strictly formal, 
but always uses intuitions, i.e. the a priori knowledge of space and 
time. Thanks to the progress of Symbolic Logic, especially as treated 
by Professor Peano, this part of the Kantian philosophy is now capable 
of a final and irrevocable refutation. By the help of ten principles 
of deduction and ten other premisses of a general logical nature 
{e.g. " implication is a relation ^^), all mathematics can be strictly and 
formally deduced ; and all the entities that occur in mathematics can 
be defined in terms of those that occur in the above twenty premisses. 



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2-6] 



Definition of Pure Mathematics 



In this statement. Mathematics includes not only Arithmetic and 
Analysis, but also Geometry, Euclidean and non-Euclidean, rational 
Dynamics, and an indefinite number of other studies still unborn or in 
their infancy. The fact that all Mathematics is Symbolic Logic is one 
of the greatest discoveries of our age ; and when this fact has been 
established, the remainder of the principles of mathematics consists in 
the analysis of Symbolic Logic itself. 

5. The general doctrine that all mathematics is deduction by 
logical principles from logical principles was strongly advocated by 
Leibniz, who urged constantly that axioms ought to be proved and 
that all except a few fundamental notions ought to be defined. But 
owing partly to a faulty logic, partly to belief in the logical necessity 
of Euclidean Geometry, he was led into hopeless errors in the endeavour 
to carry out in detail a view which, in its general outline, is now known 
to be correct*. The actual propositions of Euclid, for example, do not 
follow from the principles of logic alone ; and the perception of this fact 
led Kant to his innovations in the theory of knowledge. But since 
the growth of non-Euclidean Geometry, it has appeared that pure 
mathematics has no concern with the question whether the axioms 
and propositions of Euclid hold of actual space or not : this is a question 
for applied mathematics, to be decided, so far as any decision is possible, 
by experiment and observation. What pure mathematics asserts is merely 
that the Euclidean propositions follow from the Euclidean axioms — i.e. 
it asserts an implication: any space which has such and such properties 
has also such and such other properties. Thus, as dealt with in pure 
mathematics, the Euclidean and non-Euclidean Geometries are equally 
true: in each nothing is affirmed except implications. All propositions 
as to what actually exists, like the spaee we live in, belong to experi- 
mental or empirical science, not to mathematics ; when they belong to 
applied mathematics, they arise from giving to one or more of the 
variables in a proposition of pure mathematics some constant value 
satisfying the hypothesis, and thus enabling us, for that value of the 
variable, actually to assert both hypothesis and consequent instead of 
asserting merely the implication. We assert always in mathematics 
that if a certain assertion p is true of any entity -r, or of any set of 
entities ^r, ?/, ^, . . . , \hejx ,some other assertion q is true of those entities ; 
but we do not assert Q^^^^'^j^r q separately of our entities. We assert 

leiptions p and q^ which I shall call Jbrmal 




a relation betw^g^f^k! 
implication. 
6. ^ 

^not: 
;ic has to 

* On this su 



tuitions are not only characterized by the 

ions, but also by the fact that they contain 

e variable is one of the most difficult with 

in the present work a satisfactory theory 

outurat. La Logique de Leibniz, Paris_, 1901. 



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6 The Indejinables of Mathematics [chap, i 

as to its nature, in spite of much discussion, will hardly be found. 
For the present, I only wish to make it plain that there are variables 
in all mathematical propositions, even where at first sight they might 
seem to be absent. Elementary Arithmetic might be thought to form 
an exception : 1 + 1=2 appears neither to contain variables nor to 
assert an implication. But as a matter of fact, as will be shown in 
Part II, the true meaning of this proposition is : ^^ If ^ is one and 
y is one, and x diifers from y^ then x and y are two." And this 
proposition both contains variables and asserts an implication. We 
shall find always, in all mathematical propositions, that the words any 
or some occur ; and these words are the marks of a variable and a formal 
implication. Thus the above proposition may be expressed in the form : 
" Any unit and any other unit are two units." The typical proposition 
of mathematics is of the form "^{oc^ y^ ^? •••) implies '^{oo^ «/, ^, ...), 
whatever values oc^ «/, ^, ... may have"; where ^(<r, «/, ^, ...) and- 
i/r(^, «/, ^,...), for every set of values of x^ ?/, ^, ..., are propositions. 
It is not asserted that (j> is always true, nor yet that i|r is always true, 
but merely that, in all cases, when <^ is false as much as when ^ is true, 
i/r follows from it. 

The distinction between a variable and a constant is somewhat 
obscured by mathematical usage. It is customary, for example, to speak 
of parameters as in some sense constants, but this is a usage which 
we shall have to reject. A constant is to be something absolutely 
definite, concerning which there is no ambiguity whatever. Thus 1, ^, 
3, ^, TT, Socrates, are constants ; and so are man^ and the human race, 
past, present and future, considered collectively. Proposition, implica- 
tion, class, etc. are constants ; but a proposition, any proposition, some 
proposition, are not constants, for these phrases do not denote one 
definite object. And thus what are called parameters are simply 
variables.. Take, for example, the equation ax-\-by ■{•c = 0^ considered 
as the equation to a straight line in a plane. Here we say that x and y 
are variables, while a, 5, c are constants. But unless we are dealing 
with one absolutely particular line, say the line from a particular point 
in London to a particular point in Cambridge, our «, 5, c are not 
definite numbers, but stand for any numbers, and are thus also variables. 
And in Geometry nobody does deal with actual particular lines ; we 
always discuss any line. The point is that we collect the various 
couples x^ y into classes of classes, each class being defined as those 
couples that have a certain fixed relation to one triad (a, Z>, 6'). But 
from class to class, a^ Z>, c also vary, and are therefore properly variables. 

7. It is customary in mathematics to ri^gard our variables as 
restricted to certain classes : in Arithmetic, for instance, tliey are 
supposed to stand for numbers. But this onJy means that /)^they 
stand for numbers, they satisfy some formula, i.e, the hypothesis that 
they are numbers implies the formula. This, then, is what is reall} 



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6-8] Definition of Pure Mathematics 7 

asserted, and in this proposition it is no longer necessary that our 
variables should -be numbers : the implication holds equally when they 
are not so. Thus-, for example, the proposition " £c and i/ are numbers 
implies (^ + ^)^ = ^'^ -f 2^?/ + 7/^ '" holds equally if for ^ and «/ we substi- 
tute Socrates and Plato* : both hypothesis and consequent, in this case, 
will be false, but the implication will still be true. Thus in every 
proposition of pure mathematics, when fully stated, the variables have 
an absolutely unrestricted field: any conceivable entity may be substi- 
tuted for any one of our variables without impairing the truth of our 
proposition. 

8. We can now understand why the constants in mathematics are 
to be restricted to logical constants in the sense defined above. The 
process of transforming constants in a proposition into variables leads 
to what is called generalization, and gives us, as it were, the formal 
essence of a proposition. Mathematics is interested exclusively in types 
of propositions ; if a proposition p containing only constants be proposed, 
and for a certain one of its terms we imagine others to be successively 
substituted, the result will in general be sometimes true and sometimes 
false. Thus, for example, we have "Socrates is a man"; here we turn 
Socrates into a variable, and consider "x is a man." Some hypotheses 
as to cT, for example, "x is a Greek," insure the truth of "^ is a man"; 
thus "x is a Greek" implies "x is a man," and this holds for all values of 
X. But the statement is not one of pure mathematics, because it depends 
upon the particular nature of Greek and man. We may, however, vary 
these too, and obtain: If a and b are classes, and a is contained in 5, 
then "x is an a" implies "x is a 5." Here at last we have a proposition 
of pure mathematics, containing three variables and the constants class^ 
contained in^ and those involved in the notion of formal implications with 
variables. So long as any term in our proposition can be turned into 
a variable, our proposition can be generalized; and so long as this is 
possible, it is the business of mathematics to do it. If there are several 
chains of deduction which differ only as to the meaning of the symbols, 
so that propositions symbolically identical become capable of several 
interpretations, the proper course, mathematically, is to form the class of 
meanings which may attach to the symbols, and to assert that the 
formula in question follows from the hypothesis that the symbols belong 
to the class in question. In this way, symbols which stood for constants 
become transformed into variables, and new constants are substituted, 
consisting of classes to which the old constants belong. Cases of such 
generalization are so frequent that many will occur at once to every 
mathematician, and innumerable instances will be given in the present 
work. Whenever two sets of terms have mutual relations of the same 

■^ It is necessary to suppose arithmetical addition and multiplication defined (a^ 
may be easily done) so that the above formula remains significant when a; and y arc 
not numbers. 



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8 The Indefinahles of Mathematics [chap, i 

type, the same form of deduction will apply to both. For example, the 
mutual relations of points in a Euclidean plane are of the same type as 
those of the complex numbers; hence plane geometry, considered as a 
branch of pure mathematics, ought not to decide whether its variables 
are points or complex numbers or some other set of entities having the 
same type of mutual relations. Speaking generally, we ought to deal, 
in every branch of mathematics, with any class of entities whose mutual 
relations are of a specified type; thus the class, as well as the particular 
term considered, becomes a variable, and the only true constants are the 
types of relations and wl^at they involve. Now^ a type of relation is to 
mean, in this discussion, a class of relations characterized by the above 
formal identity of the deductions possible in regard to the various 
members of the class; and hence a type of relations, as will appear more 
fully hereafter, if not already evident, is always a class definable in 
terms of logical constants*. We may therefore define a type of relations 
as a class of relations defined by some property definable in terms of 
logical constants alone. ^ 

9. Thus pure mathematics must contain no indefinahles except 
logical constants, and consequently no premisses, or indemonstrable 
propositions, but such as are concerned exclusively with logical constants 
and with variables. It is precisely this that distinguishes pure from 
applied mathematics. In applied mathematics, results which have been 
shown by pure mathematics to follow from some hypothesis as to the 
variable are actually asserted of some constant satisfying the hypothesis 
in question. Thus terms which were variables become constant, and a 
new premiss is always required, namely: this particular entity satisfies 
the hypothesis in question. Thus for example Euclidean Geometry, as a 
branch of pure mathematics, consists wholly of propositions having the 
hypothesis "aS^ is a Euclidean space.'' If we go on to: "The space 
that exists is Euclidean," this enables us to assert of the space that exists 
the consequents of all the hypothetical constituting Euclidean Geometry, 
where now the variable S is replaced by the constant actual space. But 
by this step we pass from pure to applied mathematics. 

10. The connection of mathematics with logic, according to the 
above account, is exceedingly close. The fact that all mathematical 
constants are logical constants, and that all the premisses of mathematics 
are concerned with these, gives, I believe, the precise statement of what 
philosophers have meant in asserting that mathematics is a priori. The 
fact is that, when once the apparatus of logic has been accepted, all 
mathematics necessarily follows. The logical constants themselves are 
to be defined only by enumeration, for they are so fundamental that all 
the properties by which the class of them might be defined presuppose 

■^ One-one^ many-one^ transitive^ symmetrical^ are instances of types of relations 
with which we shall be often concerned. 



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8-10] Defi/rMion of Pure Mathematics 9 

some terms of the cl4ss. But practically, the method of discovering the 
logical constants is the analysis of symbolic logic, , which will be the 
business of the following chapters. The distinction of mathematics from 
logic is very arbitrary, but if a distinction is desired, it may be made as 
follows^ Logic consists of the premisses of mathematics, together with 
all other propositions which are concerned exclusively with logical 
constants and with variables but do not fulfil the above definition of 
mathematics (§ 1). Mathematics consists of all the consequences of the 
above premisses which assert formal implications containing variables, 
together with such of the premisses themselves as have these marks. 
Thus some of the premisses of mathematics, e.g. the principle of the 
syllogism, "if p implies g and q implies r, then p implies r,'' will 
belong to mathematics, while others, such as "implication is a relation,^** 
will belong to logic but not to mathematics. But for the desire to 
' adhere to usage, we might identify mathematics and logic, and define 
either as the class of propositions containing only variables and logical 
constants ; but respect for tradition leads me rather to adhere to the 
above distinction, while recognizing that certain propositions belong to 
both sciences. 

From what has now been said, the reader will perceive that the 
present work has to fulfil two objects, first, to show that all mathematics 
follows from sym!bolic logic, and secondly to discover, as far as possible, 
what are the principles of symbolic logic itself. The first of these objects 
will be pursued in the following Parts, while the second belongs to 
Part I. And first of all, as a preliminary to a critical analysis, it will 
be necessary to give an outline of Symbolic Logic considered simply as a 
branch of mathematics. This will occupy the following chapter. 



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CHAPTER 11. 



SYMBOLIC LOGIC. 

11. Symbolic or Formal Logic — I shall use these terms as 
synonyms — is the study of the various general types of deduction. 
The word symbolic designates the subject by an accidental characteristic, 
for the employment of mathematical symbols, here as elsewhere, is merely 
a theoretically irrelevant convenience. The syllogism in all its figures 
belongs to Symbolic Logic, and would be the whole subject if all 
deduction were syllogistic, as the scholastic tradition supposed. It is 
from the recognition of asyllogistic inferences that modern Symbolic 
Logic, from Leibniz onward, has derived the motive to progress. Since 
the publication of Boole's Laws of Thought (1854), the subject has 
been pursued with a certain vigour, and has attained to a very consider- 
able technical development*. Nevertheless, the subject achieved almost 
nothing of utility either to philosophy or to other branches of mathematics, 
until it was transformed by the new methods of Professor Peano+. 
Symbolic Logic has now become not only absolutely essential to every 
philosophical logician, but also necessary for the comprehension of 
mathematics generally, and even for the successful practice of certain 
branches of mathematics. How useful it is in practice can only be 
judged by those who have experienced the increase of power derived 
from acquiring it; its theoretical functions must be briefly set forth in 
the present chapter :|:. 



■^ By far the most complete account of the non-Peanesque methods will be found 
in the three volumes of Schroder_, Vorlesungen iiher die Algebra der Logik, Leipzig, 
1890, 1891, 1895. 

t See Formulaire de Mathematiques , Turin, 1895, with subsequent editions in 
later years ; also Revue de Mathematiques , Vol. vii. No. 1 (1900). The editions of 
the Formulaire will be quoted as F. 1895 and so on. The Revue de Mathematiques, 
which was originally the Rivista di Matematica, will be referred to as R. d. M. 

X In what follows the main outlines are due to Professor Peano, except as 
regards relations ; even in those cases where I depart from his views, the problems 
considered have been suggested to me by his works. 



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11-13] Symbolic Logic 11 

12. Symbolic Logic is essentially concerned with inference in 
general *5 and is distinguished from various special branches of mathe- 
matics mainly by its generality. Neither mathematics nor symbolic 
logic will study such special relations as (say) temporal priority, but 
mathematics will deal explicitly with the class of relations possessing 
the formal properties of temporal priority — properties which are 
summed up in the notion of continuity f. And the formal properties 
of a relation may be defined as those that can be expressed in terms 
of logical constants, or again as those which, while they are preserved, 
permit our relation to be varied without invalidating any inference in 
which the *said relation is regarded in the light of a variable. But 
■symbolic logic, in the narrower sense which is convenient, will not 
investigate what inferences are possible in respect of continuous relations 
{i.e. relations generating continuous series); this investigation belongs 
to mathematics, but is still too special for symbolic logic. What 
symbolic logic does investigate is the general rules by which inferences 
are made, and it requires a classification of relations or propositions 
only in so far as these general rules introduce particular notions. The 
particular notions which appear in the propositions of symbolic logic, 
and all others definable in terms of these notions, are the logical 
constants. The number of indefinable logical constants is not great : 
it appears, in fact, to be eight or nine. These notions alone form the 
subject-matter of the whole of mathematics : no others, except such 
as are definable in terms of the original eight or nine, occur anywhere 
in Arithmetic, Geometry, or rational Dynamics. For the technical 
«tudy of Symbolic Logic, it is convenient to take as a single indefinable 
the notion of a formal implication, i.e. of such propositions as "x is 
a man implies ^ is a mortal, for all values of ^^'—propositions whose 
general type is : " ^{x) implies i/r (^) for all values of ^,*" where <^ {x\ 
•y\r {oo\ for all values of zr, are propositions. The analysis of this notion 
of formal implication belongs to the principles of the subject, but is not 
required for its formal development. In addition to this notion, we 
require as indefinables the following : Implication between propositions 
not containing variables, the relation of a term to a class of which it 
is a member, the notion of such that, the notion of relation, and truth. 
By means of these notions, all the propositions of symbolic logic can be 
stated. 

13. The subject of Symbolic Logic consists of three parts, the 
calculus of propositions, the calculus of classes, and the calculus of 
relations. Between the first two, there is, within limits, a certain 
parallelism, which arises as follows: In any symbolic expression, the 

■^ I may as well say at once that I do not distinguish between inference and 
deduction. What is called induction appears to me to be either disguised deduction 
or a mere method of making plausible guesses 

+ See below, Part V, Chap, xxxvi. 



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12 The Indefinables of Mathematics [ohap. ii 

letters may be interpreted as classes or as propositions, and the relation 
of inclusion in the one case may be replaced by that of formal implication 
in the other. Thus, for example, in the principle of the syllogism, if 
a, &, c be classes, and a is contained in &, h in c, then a is contained in c; 
but if a^ h^ c be propositions, and a implies 5, h implies c, then a implies c. 
A great deal has been made of this duality, and in the later editions of 
the Formulaire^ Peano appears to have sacrificed logical precision to its 
preservation *". But, as a matter of fact, there are many ways in which 
the calculus of propositions differs from that of classes. Consider, 
for example, the following : "If p^ q^ r are propositions, and p implies- 
q or r^ then p implies q or p implies r.'' This proposition is true ; but 
its correlative is false, namely : " If a, &, c are classes, and a is contained 
in b or c, then a is contained in h or a is contained in c." For example, 
English people are all either men or women, but are not all men nor yet 
all women. The fact is that the duality holds for propositions asserting 
of a variable term that it belongs to a class, i.e, such propositions as 
" <37 is a man,'' provided that the implication involved be formal, i.e. one 
which holds for all values of x. But '^ x is a man'' is itself not a 
proposition at all, being neither true nor false; and it is not with such 
entities that we are concerned in the propositional calculus, but with 
genuine propositions. To continue the above illustration : It is true 
that, for all values of ^', " x is a man or a woman " either implies " -27 is a. 
man" or implies '^^ is a woman." But it is false that " x h a man or 
woman" either implies "x is a man" for all values of ^•, or implies 
'^ ^ is a woman " for all values of x. Thus the implication involved, which 
is always one of the two, is not formal, since it does not hold for all values 
of x^ being not always the same one of the two. The symbolic afSnity 
of the propositional and the class logic is, in fact, something of a snare,, 
and we have to decide which of the two we are to make fundamental. 
Mr McCoU, in an important series of papers t, has contended for the 
view that implication and propositions are more fundamental than 
inclusion and classes ; and in this opinion I agree with him. But he 
does not appear to me to realize adequately the distinction between 
genuine propositions and such as contain a real variable : thus he is led 
to speak of propositions as sometimes true and sometimes false, which 
of course is impossible with a genuine proposition. As the distinction 
involved is of very great importance, I shall dwell on it before proceeding 
further. A proposition, we may say, is anything that is true or that is 

■^ On the points where the duality breaks down^ cf. Schroder_, op. cit., Vol. ii^ 
Lecture 21. 

t Cf. '^'^The Calcuhis of Equivalent 'Statements/' Proceedings of the London 
Mathematical Society, Vol. ix and subsequent volumes ; '^'^ Symbolic Reasonings" Mind^ 
Jan. 1880^ Oct. 1897j, and Jan. 1900; '^'^LaLogique Symbolique et ses Apphcations/' 
Bibliotheque du Gongres International de Philosophie, Vol. in (Paris^ 1901). I shall in 
future quote the proceedings of the above Congress by the title Congres. 



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13,14] Symbolic Logic 13 

false. An expression such as " oc is a man''^ is therefore not a proposi- 
tion, for it is neither true nor false. If we give to x any constant value 
whatever, the expression becomes a proposition : it is thus as it were a 
schematic form standing for any one of a whole class of propositions. 
And when we say " ^ is a man implies ^ is a mortal for all values of oo^'' 
we are not asserting a single implication, but a class of implications ; 
we have now a genuine proposition, in which, though the letter oo appears, 
there is no real variable : the variable is absorbed in the same kind of 
way as the cc under the integral sign in a definite integral, so that the 
result is no longer a function of x\ Peano distinguishes a variable which 
appears in this way as apparent^ since the proposition does not depend 
upon the variable ; whereas in " <r is a man " there are different proposi- 
tions for different values of the variable, and the variable is what Peano 
calls real^. I shall speak of propositions exclusively where there is no 
real variable : where there are one or more real variables, and for all 
values of the variables the expression involved is a proposition, I shall 
call the expression a propositional function. The study of genuine 
propositions is, in my opinion, more fundamental than that of classes ; 
but the study of propositional functions appears to be strictly on a 
par with that of classes, and indeed scarcely distinguishable therefrom. 
Peano, like McColl, at first regarded propositions as more fundamental 
than classes, but he, even more definitely, considered propositional func- 
tions rather than propositions. From this criticism, Schroder is exempt: 
his second volume deals with genuine propositions, and points out their 
formal differences from classes. 

A. The Propositional Calculus. 

14. The propositional calculus is characterized by the fact that 
all its propositions have as hypothesis and as consequent the assertion of 
a material implication. Usually, the hypothesis is of the form " p ivn- 
plies^,'' etc., which (§ 16) is equivalent to the assertion that the letters 
which occur in the consequent are propositions. Thus the consequents 
consist of propositional functions wjiich are true of all propositions. 
It is important to observe that, though the letters employed are symbols 
for variables, and the consequents are true when the variables are given 
values which are propositions, these values must be genuine propositions, 
not propositional functions. The hypothesis " p is a proposition " is 
not satisfied if for p we put " ^ is a man,"' but it is satisfied if we put 
" Socrates is a man " or if we put '' ^ is a man implies ^ is a mortal for 
all values of oo.'''' Shortly, we may say that the propositions represented 
by single letters in this calculus are variables, but do not contain 
variables — in the case, that is to say, where the hypotheses of the 
propositions which the calculus asserts are satisfied. 

•^ F. 1901, p. 2. 



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14 The Indefinables of Mathematics [chap, ii 

15. Our calculus studies the relation of implication between 
propositions. This relation must be distinguished from the relation 
of formal implication, which holds between propositional functions 
when the one implies the other for all values of the variable. Formal 
implication is also involved in this calculus, but is not explicitly 
studied : we do not consider propositional functions in general, but 
only certain definite propositional functions which occur in the propo- 
sitions of our calculus. How far formal implication is definable in 
terms of implication simply, or material implication as it may be 
called, is a difficult question, which will be discussed in Chapter iii. 
What the difference is between the two, an illustration will explain. 
The fifth proposition of Euclid follows from the fourth : if the fourth 
is true, so is the fifth, while if the fifth is false, so is the fourth.' 
This is a case of material implication, for both propositions are absolute 
constants, not dependent for their meaning upon the assigning of a 
value to a variable. But each of them states a formal implication. The 
fourth states that if x and y be triangles fulfilling certain conditions, 
then X and y are triangles fulfilling certain other conditions, and that 
this implication holds for all values of x and y ; and the fifth states that 
if X is an isosceles triangle, x has the angles at the base equal. The 
formal implication involved in each of these two propositions is quite 
a different thing from the material implication holding between the 
propositions as wholes; both notions are required in the propositional 
calculus, but it is the study of material implication which specially 
distinguishes this subject, for formal implication occurs throughout the 
whole of mathematics. 

It has been customary, in treatises on logic, to confound the two 
kinds of implication, and often to be really considering the formal kind 
where the material kind only was apparently involved. For example, 
when it is said that "Socrates is a man, therefore Socrates is a mortal,'' 
Socrates i^felt as a variable: he is a type of humanity, and one, feels that 
any other man would have done as well. If, instead of therefore^ which 
implies the truth of hypothesis and consequent, we pat "Socrates is a 
man implies Socrates is a mortal,'' it appears at once that we may 
substitute not only another man, but any other entity whatever, in the 
place of Socrates. Thus although what is explicitly stated, in such a 
case, is a material implication, what is meant is a formal implication ; and 
some effbrt is needed to confine our imagination to material implication. 

16. A definition of implication is quite impossible. If p implies 
q^ then if p is true q is true, i.e. j^'s truth implies g^'s truth ; also if q is 
false p is false, i.e. g^'s falsehood implies ^'s falsehood*. Thus truth and 
falsehood give us merely new implications, not a definition of implication. 

■^ The reader is recommended to observe that the main implications in these 
statements are formal^ i.e. ^^p implies q'' formally implies '^'^^'s truth implies g's 
truth," while the subordinate implications are material. 



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16-17] Symholic Logic 15 

If p implies q^ then both are false or both true, or p is false and q true ; 
it is impossible to have q false and p true, and it is necessary to have 
q true or p false*. In fact, the assertion that q is true or p false turns 
out to be strictly equivalent to "p implies g"; but as equivalence means 
mutual implication, , this still leaves implication fundamental, and not 
definable in terms of disjunction. Disjunction, on the other hand, is 
definable in terms of implication, as we shallshbrtly see. It follows 
from the above equivalence that of any two propositions there must be 
one which implies the other, that false propositions imply all propositions, 
and true propositions are implied by all propositions. But these are 
results to be demonstrated; the premisses of our subject deal exclusively 
with rules of inference. 

It may be observed that, although implication is indefinable, 
proposition can be defined. Every proposition implies itself, and 
whatever is not a proposition implies nothing. Hence to say "p is a 
proposition'' is equivalent to saying "p implies^"; and this equivalence 
may be used' to define propositions. As the mathematical spnse of 
definition is widely different from that current among philosophers, 
it may be well to observe that,' in the mathematical sense, a new 
propositional function is said to be defined when it is stated to be 
equivalent to {i.e. to imply and be implied by) a propositional function 
which has either been accepted as indefinable or has been defined in 
terms of indefinables. The definition of entities which are not 
propositional functions is derived from such as are in ways which will 
be explained in connection with classes and relations. 

17. We require, then, in the propositional calculus, no indefinables 
except the two kinds of implication — remembering, however, that formal 
implication is a complex notion, whose analysis remains to be undertaken. 
As regards our two indefinables, we require certain indemonstrable 
propositions, which hitherto I have not succeeded in reducing to less 
than ten. Some indemonstrables there must be; and some propositions, 
such as the syllogism, must be of the number, since no demonstration 
is possible without them. But concerning others, it may be doubted 
whether they are indemonstrable or merely undemonstrated; and it 
should be observed that the method of supposing an axiom false, and 
deducing the consequences of this assumption, which has been found 
admirable in such cases as the axiom of parallels, is here not universally 
available. For all our axioms are principles of deduction; and if they 
are true, the consequences which appear to follow from the employment 
of an opposite principle will not really follow, so that arguments from 
the supposition of the falsity of an axiom are here subject to special 
fallacies. Thus the number of indemonstrable propositions may be 
capable of further reduction, and in regard to some of them I know of 

■^ I may as well state once for all that the alternatives of a disjunction will never 
be considered as mutually exclusive unless expressly said to be so. 



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16 The Indefinables of Mathematics [chap, ii 

no grounds for regarding them as indemonstrable except that they have 
hitherto remained undemonstrated. 

18. The ten axioms are the following. (1) If p implies q^ then 
p implies g^*; in other words, whatever p and q may be, " p implies g" 
is a proposition. (2) If p implies q^ then p implies p\ in other words, 
whatever implies anything is a proposition. (3) If p implies q^ then q 
implies g; in other words, whatever is implied by anything is a proposition. 
(4) A true hypothesis in an implication may be dropped, and the 
consequent asserted. This is a principle incapable of formal symbolic 
statement, and illustrating the essential limitations of formalism — a 
point to which I shall return at a later stage, t Before proceeding 
jfurther, it is desirable to define the joint assertion of two propositions, 
or what is called their logical product. This definition is highly artificial, 
and illustrates the great distinction between mathematical and philo- 
sophical definitions. It is as follows: If p implies jp, then, if q implies q^ 
pq (the logical product of p and q) means that if p implies that q implies 
r, then r is true. In other words, if ^ and q are propositions, their joint 
assertion is equivalent to saying that every proposition is true which is 
such that the first implies that the second implies it. \ We cannot, with 
formal correctness, state our definition in this shorter form, for the 
hypothesis "p and q are propositions" is already the logical product of 
"p is a proposition" and ''^ is a proposition." We can now state the 
six main principles of inference, to each of which, owing to its importance, 
a name is to be given; of these all except the last will be found in 
Peano^s accounts of the subject. (5) If p implies p and q implies q^ 
then pq implies p. This is called simplification^ and asserts merely that 
the joint assertion of two propositions implies the assertion of the first 
of the two. (6) If p implies q and q implies r, then p implies r. This 
will be called the syllogism, (7) If q implies q and r implies r, and 
if p implies that q implies r, then pq implies r. This is the principle of 
importation. In the hypothesis, we have a product of three propositions ; 
but this can of course be defined by means of the product of two. 
The principle states that if p implies that q implies r, then r follows 
from the joint assertion of p and q. For example: "If I call on so-and- 
so, then if she is at home I shall be admitted" implies "If I call on 
so-and-so and she is at home, I shall be admitted." (8) If p implies 
p and q implies ^, then, if pq implies r^ then p implies that q implies r. 
This is the converse of the preceding principle, and is called exportation f. 
The previous illustration reversed will illustrate this principle. (9) If 
p implies q and p implies r, then p implies qr: in other words, a 

■^ Note that the implications denoted by if and then, in these axioms^ are formal^ 
while those denoted by implies are materiaL 

t (7) and (8) cannot (I think) be deduced from the definition of the logical 
product^ because they are required for passing from ^^If p is a proposition^ then ^q is 
a proposition' implies etc." to ^^If p and q are propositions^ then etc." 



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17-19] Symbolic Logic 17 

proposition which impHes each of two propositions . impHes them both. 
This is called the principle of composition. (10) If j?) implies p and 
q implies g, then "'p implies q'' implies j^" implies p. This is called 
the principle of reduction; it has less self-evidence than the previous 
principles, but is equivalent to many propositions that are self-evident. 
I prefer it to these, because it is explicitly concerned, like its predecessors, 
with implication, and has the same kind of logical character as they 
have. If we remember that "p implies g" is equivalent to "q or not-p," 
we can easily convince ourselves that the above principle is true; for 
"'p implies q'' implies J9" is equivalent to "p or the denial of 'q or not- 
j^,'" ie. to "p or 'p and not g^,'" i.e. to p. But this way of persuading 
ourselves that the principle of reduction is true involves many logical 
principles which have not yet been demonstrated, and cannot be 
demonstrated except by reduction or some equivalent. The principle is 
especially useful in connection with negation. Without its help, by 
means of the iirst nine principles, we can prove the law of contradiction; 
we can prove, if p and q be propositions, that p implies not-not-p ; that 
"p implies not-^" is equivalent to "q implies not-p" and to not-pq; 
that "p implies g'" implies "not-g implies not-p"; that p implies that 
not-p implies j9; that not-p is equivalent to "p implies not-/?'\ and that 
"p implies not-g^" is equivalent to "not-not-p implies not-g.'** But we 
cannot prove without reduction or some equivalent (so far at least as 
I have been able to discover) that p or not-p must be true (the law of 
excluded middle); that every proposition is equivalent to the negation 
of some other proposition; that not-not-p implies j9; that "not-q implies 
not-p"' implies "p implies g''; that "not-p implies p>"" implies p, or that 
"p implies q""' implies "q or not-jt?.'"* Each of these assumptions is 
equivalent to the principle of reduction, and may, if we choose, be sub- 
stituted for it. Sojne of them — especially excluded middle and double 
negation — appear to have far more self-evidence. But when we have 
seen how to define disjunction and negation in terms of implication, we 
shall see that the supposed simplicity vanishes, and that, for formal 
purposes at any rate, reduction is simpler than any of the possible 
alternatives. For this reason I retain it among my premisses in 
preference to more usual and more superficially obvious propositions. 
19. Disjunction or logical addition is defined as follows: "p or g" 
is equivalent to "'p implies g' implies g." It is easy to persuade 
ourselves of this equivalence, by remembering that a false proposition 
implies every other; for if p is false, p does imply g, and therefore, 
if "p implies g" implies g, it follows that g is true. But this argument 
again uses principles which have not yet been demonstrated, and is 
merely designed to elucidate the definition by anticipation. From this 
definition, by the help of reduction, we can prove that "p or g'' is 
equivalent to "g or jy." An alternative definition, deducible from the 
above, is : "Any proposition implied by p and implied by g is true," or, 
R. 2 



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18 The Indejinables of Mathematics [chap, il 

in other words, " ' p implies s'' and 'q implies s'' together imply ^5 whatever 
s may be."" Hence we proceed to the definition of negation: not-p is 
equivalent to the assertion that 'p implies all propositions, i.e. that 
"r implies r" implies "p implies r*" whatever r may be*. From this 
point we can prove the laws of contradiction and excluded middle and 
double negation, and establish all the formal properties of logical 
multiplication and addition— the associative, commutative and distributive 
laws. Thus the logic 6i propositions is now complete. 

Philosophers will object to the above definitions of disjunction and 
negation on the ground that what we mean by these notions is some- 
thing quite distinct from what the definitions assign as their meanings, 
and that the equivalences stated in the definitions are, as a matter of 
fact, significant propositions, not mere indications as to the way in 
which symbols are going to be used. Such an objection is, I think, well- 
founded, if the above account is advocated as giving the true philosophic 
analysis of the matter. But where a purely formal purpose is to be 
served, any equivalence in which a certain notion appears on one side 
but not on the other will do for a definition. And the advantage of 
having before our minds a strictly formal development is that it pro- 
vides the data for philosophical analysis in a more definite shape than 
would be otherwise possible. Criticism of the procedure of formal logic, 
therefore, will be best postponed until the present brief account has been 
brought to an end. 

B. The Calculus of Classes. 

20. In this calculus there are very much fewer new primitive pro- 
positions — in fact, two seem sufficient — but there are much greater 
difficulties in the way of non-symbolic exposition of the ideas embedded 
in our symbolism. These difficulties, as far as possible, will be postponed 
to later chapters. For the present, I shall try to make an exposition 
which is to be as straightforward and simple as possible. 

The calculus of classes may be developed by regarding as fundamental 
the notion of class^ and also the relation of a member of a class to its 
class. , This method is adopted by Professor Peano, and is perhaps more 
philosophically correct than a different method which, for formal pur- 
poses, I have found more convenient. In this method we still take as 

^ The principle that false propositions imply all propositions solves Lewis 
Carroll's logical paradox in Mind, N. S. No. 11 (1894). The assertion made in that 
paradox is that^ if p, q, r he propositions^ and q implies r, while p implies that 
q implies not-r^ then p must he false^ on the supposed ground that ^'q implies r" and 
[' q implies not-r " are incompatible. But in virtue of our definition of negation^ if 
q he false both these implications will hold : the two together^ in fact^ what- 
ever proposition r may be^ are equivalent to not-^. Thus the only inference 
warranted by Lewis Carroll's premisses is that if p be true^ q must be false^ i.e. that 
p implies not-? ; and this is the conclusion^ oddly enough^ which common sense would 
have drawn in the particular case which he discusses. 



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19-22] Symbolic Logic 19 

fundamental the relation (which, following Peano, I shall denote by e) 
of an individual to a class to which it belongs, ix. the relation of Socrates 
to the human race which is expressed by saying that Socrates is a man. 
In addition to this, we take as indefinables the notion of a propositional 
function and the notion of such that. It is these three notions that 
characterize the class-calculus. Something must be said in explanation 
of each of them. 

21. The insistence on the distinction between e and the relation of 
whole and part between classes is due to Peano, and is of very great 
importance to the whole technical development and the whole of the 
applications to mathematics. In the scholastic doctrine of the syllogism, 
and in all previous symbolic logic, the two relations are confounded, 
except in the work of Frege*. The distinction is the same as that 
between the relation of individual to species and that of species to 
genus, between the relation of Socrates to the class of Greeks and the 
relation of Greeks to men. On the philosophical nature of this distinc- 
tion I shall enlarge when I come to deal critically with the nature of 
classes; for the present it is enough to observe that the relation of 
whole and part is transitive, while e is not so : we have Socrates is a 
a man, and men are a class, but not Socrates is a class. It is to be 
observed that the class must be distinguished from the class-concept 
or predicate by which it is to be defined : thus men are a class, while 
mian is a class-concept. The relation e must be regarded as holding 
between Socrates and men considered collectively, not between Socrates 
and vian. I shall return to this point in Chapter vi. Peano holds 
that all propositional functions containing only a single variable are 
capable of expression in the form " oc is an a^"" where a is a constant 
class ; but this view we shall find reason to doubt. 

22. The next fundamental notion is that of a propositional func- 
tion. Although propositional functions occur in the calculus of pro- 
positions, they are there each defined as it occurs, so that the general 
notion is not required. But in the class-calculus it is necessary to intro- 
duce the general notion explicitly. Peano does not require it, owing to 
his assumption that the form "oc is an a" is general for one variable, and 
that extensions of the same form are available for any number of 
variables. But we must avoid this assumption, and must therefore 
introduce the notion of a propositional function. We may explain (but 
not define) this notion as follows : ^^ is a propositional function if, for 
every value of ^, c^x is a proposition, determinate when x is given. 
Thus " ^ is a man '' is a propositional function. In any proposition, how- 
ever complicated, which contains no real variables, we may imagine one 
of the terms, not a verb or adjective, to be replaced by other terms : instead 
of " Socrates is a man '" we may put " Plato is a man," " the number S 

■^ See his Begriffsschrift, Halle^ 1879^ and Grundgesetze der Arithmetik, Jena, 1893;, 
p. 2. 

2—2 



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20 The Indejinables of Mathematics [chap, ii 

is a man,'' and so on*. Thus we get successive propositions all agreeing 
except as to the one variable term. Putting x for the variable term, 
"x is a man '\ expresses the type of all such propositions. A pro- 
positional function in general will be true for some values of the variable 
and false for others. The instances where it is true for all values of the 
variable, so far as they are known to me, all express implications, such as 
" ^ is a man implies ^ is a mortal"; but I know of no a priori reason for 
asserting that no other propositional functions are true for all values of 
the variable. 

23. This brings me to the notion of such that. The values of x 
which render a propositional function (^x true are like the roots of an 
equation — indeed the latter are a particular case of the former — and we 
may consider all the values of x which are such that (j)X is true. In general, 
these values form a class^ and in fact a class may be defined as all 
the terms satisfying some propositional function. There is, however, 
some limitation required in this statement, though I have not been able to 
discover precisely what the limitation is. This results from a certain 
contradiction which I shall discuss at length at a later stage (Chap. x). 
The reasons for defining class in this way are, that we require to provide 
for the null-class, which prevents our defining a class as a term to 
which some other has the relation ^, and that we wish to be able 
to define classes by relations, i.e. all the terms which have to other 
terms the relation R are to form a class, and such cases require somewhat 
complicated propositional functions. 

24. With regard to these three fundamental notions, we require 
two primitive propositions. The first asserts that if x belongs to the 
class of terms satisfying a propositional function (px^ then (px is true. 
The second asserts that if cpx and yjrx are equivalent propositions for all 
values of x, then the class of ^'s such that (j)X is true is identical with 
the class of ^'s such that 'yfrx is true. Identity, which occurs here, is 
defined as follows : x is identical with y ii y belongs to every class to 
which ^belongs, in other words, if ".r is a t;'' implies "^5/ is a z^" for 
all values of u. With regard to the primitive proposition itself, it is to 
be observed that it decides in favour of an extensional view of classes. 
Two class-concepts need not be identical when their extensions are so : 
man sjiAfeatheiiess biped are by no means identical, and no more are even 
p7ime and integer between 1 and B. These are class-concepts, and if our 
axiom is to hold, it must not be of these that we are to speak in dealing 
with classes. We must be concerned with the actual assemblage of 
terms, not with any concept denoting that assemblage. For mathe- 
matical purposes, this is quite essential. Consider, for example, the 
problem as to how many combinations can be formed of a given set 

■^ Verbs and adjectives occurring as such are distinguished by the fact that^ if 
they be taken as variable;, the resulting function is only a proposition for some values 
of the variable, i.e. for such as are verbs or adjectives respectively. See Chap. iv. 



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22-25] Symbolic Logic 21 

of terms taken any number at a time, i.e. as to how many classes are 
contained in a given class. If distinct classes may have the same ex- 
tension, this problem becomes utterly indeterminate. And certainly 
common usage would regard a class as determined when all its terms are 
given. The extensional view of classes, in some form, is thus essential to 
Symbolic Logic and to mathematics, and its necessity is expressed in the 
above axiom. But the axiom itself is not employed until we come to 
Arithmetic ; at least it need not be employed, if we choose to distinguish 
the equality of classes, which is defined as mutual inclusion, from the 
identity of individuals. Formally, the two are totally distinct : identity 
is defined as above, equality of a and b is defined by the equivalence of 
" oc is an a ^' and " ^ is a 5 '^ for all values of oo. 

25. Most of the propositions of the class-calculus are easily 
deduced from those of the propositional calculus. The logical product 
or common part of two classes a and h is the class of ^''s such that the 
logical product of " x is an a'''' and "x is a V is true. Similarly we define 
the logical sum of two classes {a or 5), and the negation of a class (not-a). 
A new idea is introduced by the logical product and sum of a class of 
classes. If A; is a class of classes, its logical product is the class of terms be- 
longing to each of the classes of A;, i.e. the class of terms oo such that " u 
is a ^ " implies " ^ is a u " for all values of u. The logical sum is the class 
which is contained in every class in which every class of the class h is 
contained, i.e. the class of terms oo such that, if " ^^ is a A;'" implies "ii is 
contained in c" for all values of w, then, for all values of ^, ^ is a c. 
And we say that a class a is contained in a class h when " oo is an a'''' 
implies " ^ is a 5 " for all values of oc. In like manner with the above 
we may define the product and sum of a class of propositions. Another 
very important notion is what is called the existence of a class — a word 
which must not be supposed to mean what existence means in philosophy. 
A class is said to exist when it has at least one term. A formal defini- 
tion is as follows : a is an existent class wh^n and only when any 
proposition is true provided "ir is an a"' always implies it whatever value 
we may give to x. It must be understood that the proposition implied 
must be a genuine proposition, not a propositional function of x. A 
class a exists when the logical sum of all propositions of the form " x is 
an a " is true, i.e. when not all such propositions are false. 

It is important to understand clearly the manner in which pro- 
positions in the class-calculus are obtained from those in the pro- 
positional calculus. Consider, for example, the syllogism. We have 
"p implies gr'*' and "q implies r" imply '^ p implies r.*" Now put " x is 
an «,"" "x is a h^'' "x is a c" forp, q^ r, where x must have some definite 
value, but it is not necessary to decide what value. We then find that 
if, for the value of x in question, ^ is an « implies /r is a 5, and xi^^h 
implies ^ is a c, then ^r is an a implies x is a c. Since the value of x is 
irrelevant, we may vary x^ and thus we find that if a is contained in 6, 



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22 The Indefinables of Mathematics [ohap. il 

and h in c, then a is contained in c. This is the class-syllogism. But in 
applying this process it is necessary to employ the utmost caution, 
if fallacies are to be successfully avoided. In this connection it will 
be instructive to examine a point upon which a dispute has arisen 
between Schroder and Mr McColl*. Schroder asserts that if jt?, q^ r are 
propositions, "pq implies r" is equivalent to the disjunction "p implies r 
or q implies r.'' Mr McColl admits that the disjunction implies the 
other, but denies the converse implication. The reason for the diver- 
gence is, that Schroder is thinking of propositions and material im- 
plication, while Mr McColl is thinking of propositional functions and 
formal implication. As regards propositions, the truth of the principle 
may be easily made plain by the following considerations. If pq implies 
r, then, if either p or q be false, the one of them which is false implies r, 
because false propositions imply all propositions. But if both be true, 
pq is true, and therefore r is true, and therefore jo implies r and q im- 
plies r, because true propositions are implied by every proposition. 
Thus in any case, one at least of the propositions p and q must 
imply r. (This is not a proof, but an elucidation.) But Mr McColl 
objects: Suppose j9 and q to be mutually contradictory, and r to be the 
null proposition, then pq implies r but neither p nor q implies r. Here 
we are dealing with propositional functions and formal implication. A 
propositional function is said to be niill when it is false for all values of 
X ; and the class of oo\ satisfying the function is called the null-class, 
being in fact a class of no terms. Either the function or the class, 
following Peano, I shall denote by A. Now let our r be replaced by A, 
our p by ^^, and our q by not-<^<27, where c^x is any propositional function. 
Then pq is false for all values of ^, and therefore implies A. But it is 
not in general the case that ^oo is always false, nor yet that not-^<3? is always 
false; hence neither always implies A. Thus the above formula can only 
be truly interpreted in the propositional calculus : in the class-calculus 
it is false. This may be easily rendered obvious by the following 
considerations : Let (^^, i/r<r, ')(X be three propositional functions. Then 
" (j^x . -v/rzr implies ;^^''" implies, for all values of x^ that either (^x implies 
'Xoo or '\\rx implies ')(x. But it does not imply that either ^^ implies ')(x 
for all values of x^ or y^rx implies ')(^x for all values oi x. The disjunction 
is what I shall call a variable disjunction, as opposed to a constant one: 
that is, in some cases one alternative is true, in others the other, whereas 
in a constant disjunction there is one of the alternatives (though it is not 
stated which) that is always true. Wherever disjunctions occur in regard 
to propositional functions, they will only be transformable into statements 
in the class-calculus in cases where the disjunction is constant. This is 
a point which is both important in itself and instructive in its bearings. 
Another way of stating the matter is this: In the proposition: If 

* Schroder;, Algebra der Logik, Vol. ii, pp. 258-9; McColl^, ^^ Calculus of 
Equivalent Statements," fiftli paper, Proc. Lond. Math. Soc. Vol. xxviii, p. 182. 



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25-27] Symbolic Logic 23 

^ce . -^x implies ^^^ then either (^oo implies ')(oc or i/r«r implies ')(po^ the 
implication indicated by if and then is formal, while the subordinate 
implications are material; hence the subordinate implications do not 
lead to the inclusion of one class in another, which results only from 
formal implication. 

The formal laws of addition, multiplication, tautology and negation 
are the same as regards classes and propositions. The law of tautology 
states that no change is made when a class or proposition is added to or 
multiplied by itself. A new feature of the class -calculus is the null-class, 
or class having no terms. This may be defined as the class of terms that 
belong to every class, as the class which does not exist (in the sense 
defined above), as the class which is contained in every class, as the 
class A which is such that the propositional function "^ is a A'^ is false 
for all values of zr, or as the class of ^'s satisfying any propositional 
function ^x which is false for all values of x. All these definitions are 
easily shown to be equivalent. 

26. Some important points arise in connection with the theory of 
identity. We have already defined two terms as identical when the 
second belongs to every class to which the first belongs. It is easy to 
show that this definition is symmetrical, and that identity is transitive 
and reflexive {i.e. if x and ?/, y and z are identical, so are x and z ; and 
whatever x may be, x is identical with x). Diversity is defined as the 
negation of identity. If x be any term, it is necessary to distinguish 
from X the class whose only member is x : this may be defined as the 
class of terms which are identical with x. The necessity for this 
distinction, which results primarily from purely formal considerations, 
was discovered by Peano ; I shall return to it at a later stage. Thus 
the class of even primes is not to be identified with the number % and 
the class of numbers which are the sum of 1 and 2 is not to be identified 
with 3. In what, philosophically speaking, the difference consists, is a 
point to be considered in Chapter vi. 

C. The Calculus of Relations. 

27. The calculus of relations is a more modern subject than the 
calculus of classes. Although a few hints for it are to be found in 
De Morgan*, the subject was first developed by C. S. Peircef. A careful 
analysis of mathematical reasoning shows (as we shall find in the course 
of the present work) that types of relations are the true subject-matter 
discussed, however a bad phraseology may disguise this fact ; hence the 
logic of relations has a more immediate bearing on mathematics than 

■^, Gamb. Phil. Trans. Vol. x^ ^'^On the Syllogism_, No. iv, and on the Logic of 
Relations." Cf. ih.Yol. ix, p. 104; also hm Formal Logic (London, 1847), p. -50. 

t See especially his articles on the Algehra of Logic, American Journal of 
Mathematics, Vols, iii and vii. The subject is treated at length by C. S. Peirce's 
methods in Schroder, op. cit., Vol. in. 



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24 The Indefinahles of Mathematics [chap, il 

that of classes or propositions, and any theoretically correct and adequate 
expression of mathematical truths is only possible by its means. Peirce 
and Schroder have realized the great importance of the subject, but 
unfortunately their methods, being based, not on Peano, but on the 
older Symbolic Logic derived (with modifications) from Boole, are so 
cumbrous and difficult that most of the applications which ought to be 
made are practically not feasible. In addition to the defects of the old 
Symbolic Logic, their method suffers technically (whether philosophically 
or not I do not at present discuss) from the fact that they regard a 
relation essentially as a qlass of couples, thus requiring elaborate 
formulae of summation for dealing with single relations. This view is 
derived, I think, probably unconsciously, from a philosophical error : it 
has always been customary to suppose relational propositions less 
ultimate than class-propositions (or subject-predicate propositions, with 
which class-propositions ^are habitually confounded), and this has led 
to a desire to treat relations as a kind of classes. However this may 
be, it was certainly from the opposite philosophical belief, which I 
derived from my friend Mr G. E. Moore*, that I was led to a different 
formal treatment of relations. This treatment, whether more philo- 
sophically correct or not, is certainly far more convenient and far more 
powerful as an engine of discovery in actual mathematics f. 

28. If i? be a relation, we express by ocRy the propositional function 
" X has the relation R to y^ We require a primitive {i.e. indemonstrable) 
proposition to the effect that xRy is a proposition for all values of x 
and y. We then have to consider the following classes : The class of 
terms which have the relation R to some term or other, which I call the 
class of referents with respect to R ; and the class of terms to which --^ 
some term has the relation J?, which I call the class of relata. Thus if 
R be paternity, the referents will be fathers and the relata will be 
children. We have also to consider the corresponding classes with 
respect to particular terms or classes of terms : so-and-so's children, or 
the children of Londoners, afford illustrations. 

The intensional view of relations here advocated leads to the result 
that two relations may have the same extension without being identical. 
Two relations i?, R are said to be equal or equivalent, or to have the 
same extension, when xRy implies and is implied by xR'y for all values 
of X and y. But there is no need here of a primitive proposition, as 
there was in the case of classes, in order to obtain a relation which is 
determinate when the extension is determinate. We may replace a 
relation R by the logical sum or product of the class of relations 
equivalent to i?, i.e, by the assertion of some or of all such relations ; 
and this is identical with the logical sum or product of the class of 
relations equivalent to R\ if R be equivalent to R. Here we use 

^ See his article ^' On the Nature of Judgment/* Mind, N. S. No. 30. 
+ See my articles in R. d. M, Vol. vii^ No. 2 and subsequent numbers. 



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27-29] Symbolic Logic 26 

the identity of two classes, which results from the primitive proposition 
as to identity of classes, to establish the identity of two relations — 
a procedure which could not have been applied to classes themselves 
without a vicious circle. 

A primitive proposition in regard to relations is that every relation 
has a converse, ix, that, if R be any relation, there is a relation R such 
that xRy is equivalent to yR'x for all values of x and y. Following 

Schroder, I shall denote the converse of R by R. Greater and less, 
before and after, implying and implied by, are mutually converse 
relations. With some relations, such as identity, diversity, equality, 
inequality, the converse is the same as the original relation: such 
relations are called symmetrical. When the converse is incompatible 
with the original relation, as in such cases as greater and less, I call the, 
relation asymmetrical \ in intermediate cases, not-symmetrical. 

The most important of the primitive propositions in this subject is 
that between any two terms there is a relqition not holding between any 
two other terms. This is analogous to the principle that any term is 
the only member of some class; but whereas that could be proved, 
owing to the extensional view of classes, this principle, so far as I can 
discover, is incapable of proof. .In this point, the extensional view of 
relations has an advantage; but the advantage appears to me to be 
outweighed by other considerations. When relations are considered 
intensionally, it may seem possible to d mbt whether the above principle 
is true at all. It will, however, be gene!^ally admitted that, of any two 
terms, some propositional function is true which is not true of a certain 
given different pair of terms. If this be admitted, the above principle 
follows by considering the logical product of all the relations that hold 
between our first pair of terms. Thus 'he above principle may be 
replaced by the following, which is equivalent to it : If wRy implies 
w'Ry\ whatever R may be, so long as R is a relation, then x and x^ 
y and y' are respectively identical. But this principle introduces a 
logical difficulty from which-^^e have been hitherto exempt, namely a 
variable with a restricted field ; for unless R is a relation, xRy is not a 
proposition at all, true or false, and thus R.it would seem, cannot take 
all values, but only such as are relations. I shall return to the discussion 
of this point at a later stage. 

29. Other assui^nptions required are that the negation of a. relation 
is a relation, and that the logical product of a class of relations {i.e. the 
assertion of all of them simultaneously) is a relation. Also the relative 
product of two relations must be a relation. The relative product of two 
relations J?, S is the relation which holds between x and z whenever 
there is a term y to which x has i the relation R and which has to z the 
relation S. Thus the relation of a niaternal grandfather to his grandson 
is the relative product of father and mother ; that of a paternal grand- 
mother to her grandson is the relative product of mother and father; 



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26 The Indefinables of Mathematics [chap, ir 

that of grandparent to grandchild is the relative product of parent and 
parent. The relative product, as these instances show, is not in general 
commutative, and does not in general obey the law of tautology. The 
relative product is a notion of very great importance. Since it does not 
obey the law of tautology, it leads to powers of relations : the square of 
the relation of parent and child is the relation of grandparent and 
grandchild, and so on. Peirce and Schroder consider also what they call 
the relative sum of two relations R and aS", which holds between oo and z^ 
when, if y be any other term whatever, either oo has to y the relation i?, 
or y has to z the relation aS". This is a complicated notion, which I have 
found no occasion to employ, and which is introduced only in order to 
preserve the duality of addition and multiplication. This duality has a 
certain technical charm when the subject is considered as an independent 
branch of mathematics ; but when it is considered solely in relation to 
the principles of mathematics, the duality in question appears devoid of 
all philosophical importance. 

30. Mathematics requires, so far as I know, only two other 
primitive propositions, the one that material implication is a relation,, 
the other that e (the relation of a term to a class to which it belongs) is- 
a relation*. We can now develop the whole of mathematics without 
further assumptions or indefinables. Certain propositions in the logic 
of relations deserve to be mentioned, since they are important, and it 
might be doubted whether they were capable of formal proof. If ^, v 
be any two classes, there is a relation R the assertion of which between 
any two terms x and y is equivalent to the assertion that oo belongs to u 
and y to v. If u be any class \v^hich is not null, there is a relation which 
all its terms have to it, and which holds for no other pairs of terms. If 
R be any relation, and u any class contained in the class of referents 
with respect to R^ there is u relation which has u for the class of its 
referents, and is equivalent %o R throughout that class ; this relation is 
the same as R where it holds, but has a more restricted domain. (I use 
domain as synonymous with class of referents.) From this point onwards,, 
the development of the subject is technical : special types of relations are 
considered, and special branches of mathematics result. 

D. Peano'^s Symbolic Logic. 

31. So much of the above brief outline of Symbolic Logic is 
inspired by Peano, that it seems desirable to discuss his work explicitly,, 
justifying by criticism the points in which 1 have departed from him. 

The question as to which of the notions of symbolic logic are to be 
taken as indefinable, and which of the propositions as indemonstrable,! 
is, as Professor Peano has insisted f, to some extent arbitrary. But it is 

■^ There is a difficulty in regard to this primitive proposition, discussed in § § 53^^ 
94 below. 

+ E.g. F. 1901, p. 6 ; i^. 1897, Part I, pp. 62-3. 



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29-32] Syvibolic Logic 27 

important to establish all the mutual relations of the simpler notions 
of logic, and to examine the consequence of taking various notions as 
indefinable. It is necessary to realize that definition, in mathematics, 
does not mean, as in philosophy, an analysis of the idea to be defined 
into constituent ideas. This notion, in any case, is only applicable to 
concepts, whereas in mathematics it is possible to define terms which 
are not concepts*. Thus also many notions are defined by symbolic 
logic which are not capable of philosophical definition, since they are 
simple and unanalyzable. Mathematical definition consists in pointing 
out a fixed relation to a fixed term, of which one term only is capable: 
this term is then defined by means of the fixed relation and the fixed 
term. The point in which this differs from philosophical definition 
may be elucidated by the remark that the mathematical definition does 
not point out the term in question, and that only what may be called 
philosophical insight reveals which it is among all the terms there are. 
This is due to the fact that the term is defined by a concept which 
denotes it unambiguously, not by actually mentioning the term denoted. 
What is meant by denoting^ as well as the different Avays of denoting, 
must be accepted as primitive ideas in any symbolic logic f: in this 
respect, the order adopted seems not in any degree arbitrary. 

32. For the sake of definiteness, let us now examine some one 
of Professor Peano's expositions of the subject. In his later expositions J 
he has abandoned the attempt to distinguish clearly certain ideas and 
propositions as primitive, probably because of the realization that any 
such distinction is largely arbitrary. But the distinction appears useful, 
as introducing greater definiteness, and as showing that a certain set 
of primitive ideas and propositions are sufficient ; so far from being 
abandoned, it ought rather to be made in every possible way. I shall, 
therefore, in what follows, expound one of his earlier expositions, that 
of 1897§. 

The primitive notions with which Peano starts are the following: 
Class, the relation of an individual to a class of which it is a member, 
the notion of a term, implication where both propositions contain the 
same variables, i.e. formal implication, the simultaneous affirmation of 
two propositions, the notion of definition, and the negation of a pro- 
position. From these notions, together with the division of a complex 
proposition into parts, Peano professes to deduce all symbolic logic by 
means of certain primitive propositions. Let us examine the deduction 
in outline. 

We may observe, to begin with, that the simultaneous affirmation 
of two propositions might seem, at first sight, not enough to take as a 
primitive idea. For although this can be extended, by successive steps, 
to the simultaneous affirmation of any finite number of propositions, 

"^ See Chap. iv. f See Chap. v. 

+ F. 1901 and R. d. M. Vol. vii, No. 1 (1900). § F. 1897, Part I. 



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28 The Indefinables of Matheinatics [chap. li 

yet this is not all that is wanted ; we require to be able to affirm 
simultaneously all the propositions of any class, finite or infinite. But 
the simultaneous assertion of a class of propositions, oddly enough, is 
much easier to define than that of two propositions (see § 34, (3)). If ^ 
be a class of propositions, their simultaneous affirmation is the assertion 
that "j9 is a A; " implies p. If this holds, all propositions of the class are 
true ; if it fails, one at least must be false. We have seen that the 
logical product of two propositions can be defined in a highly artificial 
manner ; but it might almost as well be taken as indefinable, since no 
further property can be proved by means of the definition. We may 
observe, also, that formal and material implication are combined by 
Peano into one primitive idea, whereas they ought to be kept separate. 

33. Before giving any primitive propositions, Peano proceeds to 
some definitions. (1) If a is a class, " x and y are a\ " is to mean 
" X is an a and y is an a.'''' (2) If a and h are classes, " every a is a Z)*" 
means " x is an a implies that ^ is a Z>." If we accept formal implication 
as a primitive notion, this definition seems unobjectionable ; but it may 
well be held that the relation of inclusion between classes is simpler than 
formal implication, and should not be defined by its means. This is a 
difficult question, which I reserve for subsequent discussion. A formal 
implication appears to be the assertion of a whole class of material 
implications. The complication introduced at this point arises from 
the nature of the variable, a point which Peano, though he has done 
very much to show its importance, appears not to have himself suffi- 
ciently considered. The notion of one proposition containing a variable 
implying another such proposition, which he takes as primitive, is 
complex, and should therefore be separated into its constituents ; from 
this separation arises the necessity of considering the simultaneous 
affirmation of a whole class of propositions before interpreting such 
a proposition as " ^ is an <^ implies that ^ is a hP'' (3) We come next 
to a perfectly worthless definition, which has been since abandoned*. 
This is the definition of such that. The ^''s such that x is an a, we are 
told, are to mean the class a. But this only gives the meaning of such 
that when placed before a proposition of the type " x is an a." Now 
it is often necessary to consider an x such that some proposition is true 
of it, where this proposition is not of the form " x is an <^." Peano holds 
(though he does not lay it down as an axiom) that every proposition 
containing only one variable is reducible to the form " x is an af." 
But we shall see (Chap, x) that at least one such proposition is not 
reducible to this form. And in any case, the only utility of such that 
is to effect the reduction, which cannot therefore be assumed to be 
already effected without it. The fact is that such that contains a primi- 
tive idea, but one which it is not easy clearly to disengage from other ideas. 

■^ In consequence of the criticisms of Padoa^ R. d, M. Vol. vi^ p. 112. 
t R, d, M. Vol. VII, No. 1, p. 25 ; F. 1901, p. 21, § 2, Prop. 4. 0, Note. 



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32-34] Symbolic Logic 29 

In order to grasp the meaning of such that^ it is necessary to observe, 
first of all, that what Peano and mathematicians generally call one 
proposition containing a variable is really, if the variable is apparent, 
the conjunction of a certain class of propositions defined by some 
constancy of form; while if the variable is real, so that we have a 
propositional function, there is not a proposition at all, but merely 
a kind of schematic representation of any proposition of a certain type: 
"The sum of the angles of a triangle is two right angles," for example, 
when stated by means of a variable, becomes : Let x be a triangle ; then 
the sum of the angles of oc is two right angles. This expresses the 
conjunction of all the propositions in which it is said of particular 
definite entities that if they are triangles, the sum of their angles is 
two right angles. But a propositional function, where the variable is 
real, represents any proposition of a certain form, not all such proposi- 
tions (see §§ 59-62). There is, for each propositional function, an 
indefinable relation between propositions and entities, which may be 
expressed by saying that all the propositions have the same form, 
but different entities enter into them. It is this that gives rise to 
propositional functions. Given, for example, a constant relation and 
a constant term, there is a one-one correspondence between the propo- 
sitions asserting that various terms have the said relation to the said 
term, and the various terms which occur in these propositions. It is 
this notion which is requisite for the comprehension of such that. Let 
<27 be a variable whose values form the class <x, and let f (oc) be a one- 
valued function of w which is a true proposition for all values of x within 
the class a, and which is false for all other values of x. Then the terms 
of a are the class of terms such thatf{x) is a true proposition. This 
gives an explanation of such that. But it must always be remembered 
that the appearance of having one proposition f{x) satisfied by a 
number of values of x is fallacious : f{x) is not a proposition at all, 
but a propositional function. What is fundamental is the relation of 
various propositions of given form to the various terms entering 
severally into them as arguments or values of the variable ; this 
relation is equally required for interpreting the propositional function 
/'(^) and the notion such that., but is itself ultimate and inexplicable. 
(4) We come next to the definition of the logical product, or 
common part, of two classes. If a and h be two classes, their common 
part consists of the class of terms x such that ^ is an « and x is a h. 
Here already, as Padoa points out {loc. cit.)^ it is necessary to extend the 
meaning of such that beyond the case where our . proposition asserts 
membership of a class, since it is only by means of the definition that 
the common part is shown to be a class. 

34. The remainder of the definitions preceding the primitive 
propositions are less important, and may be passed over. Of the 
primitive propositions, some appear to be merely concerned with the 



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30 The Indefinahles of Mathematics [chap, n 

symbolism, and not to express any real properties of what is symbolized ; 
others, on the contrary, are of high logical importance. 

(1) The first of Peano's axioms is "every class is contained in 
itself." This is equivalent to " every proposition implies itself.'' There 
seems no way of evading this axiom, which is equivalent to the law of 
identity, except the method adopted above, of using self-implication 
to define propositions. (2) Next we have the axiom that the product 
of two classes is a class. This ought to have been stated, as ought also 
the definition of the logical product, for a class of classes ; for when 
stated for only two classes, it cannot be extended to the logical product 
of an infinite class of classes. If class is taken as indefinable, it is a 
genuine axiom, which is very necessary to reasoning. But it might 
perhaps be somewhat generalized by an axiom concerning the terms 
satisfying propositions of a given form : e.g. " the terms having one 
or more given relations to one or more given terms form a class." 
In Section B, above, the axiom was wholly evaded by using a generalizfed 
form of the axiom as the definition of class. (3) We have next two 
axioms which are really only one, and appear distinct only because Peano 
defines the common part of two classes instead of the common part of a 
class of classes. These two axioms state that, if a^ h be classes, their logical 
product, ah^ is contained in a and is contained in h. These appear as 
different axioms, because, as far as the symbolism shows, ah might be 
different from ha. It is one of the defects of most symbolisms that they 
give an order to terms which intrinsically have none, or at least none 
that is relevant. So in this case : if ^ be a class of classes, the logical 
product of K consists of all terms belonging to every class that belongs 
to K. With this definition, it becomes at once evident that no order 
of the terms of K is involved. Hence if K has only two terms, a and h^ 
it is indifferent whether we represent the logical product of K by ah 
or by ha^ since the order exists only in the symbols, not in what is 
symbolized. It is to be observed that the corresponding axiom with 
regard to propositions is, that the simultaneous assertion of a class of 
propositions implies any proposition of the class ; and this is perhaps 
the best form of the axiom. Nevertheless, though an axiom is not 
required, it is necessary, here as elsewhere, to have a means of connecting 
the case where we start from a class of classes or of propositions or of 
relations with the case where the class results from enumeration of its 
terms. Thus although no order is involved in the product of a class of 
propositions, there is an order in the product of two definite proposi- 
tions ^, g, and it is significant to assert that the products pq and qp are 
equivalent. But this can be proved by means of the axioms with which 
we began the calculus of propositions (§ 18). It is to be observed that 
this proof is prior to the proof that the class whose terms are p and q is 
identical with ' the class whose terms are q and p. (4) We have next 
two forms of syllogism, both primitive propositions. The first asserts 



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34, 35] Symbolic Logic 31 

that, if a, 5, c be classes, and a is contained in 6, and ^ is an a^ then iT is 
a & ; the second asserts that if a, 6, c be classes, and a is contained in Z;, 
h in c, then a is contained in c. It is one of the greatest of Peano"'s 
merits to have clearly distinguished the relation of the individual to its 
class from the relation of inclusion between classes. The difference is 
exceedingly fundamental : the former relation is the simplest and most 
essential of all relations, the latter a complicated relation derived from 
logical implication. It results from the distinction that the syllogism 
in Barbara has two forms, usually confounded : the one the time-honoured 
assertion that Socrates is a man, and therefore mortal, the other the 
assertion that Greeks are men, and therefore mortal. These two forms 
are stated by Peano's axioms. It is to be observed that, in virtue of the 
definition of what is meant by one class being contained in another, 
the first form results from the axiom that, if p, g, r be propositions, and 
p implies that q implies r, then the product of p and q implies r. This 
axiom is now substituted by Peano for the first form of the syllogism * : 
it is more general and cannot be deduced from the said form. The 
second form of the syllogism, when applied to propositions instead of 
classes, asserts that implication is transitive. This principle is, of course, 
the very life of all chains of reasoning. (5) We have next a principle 
of reasoning which Peano calls composition : this asserts that if a is 
contained in h and also in c, then it is contained in the common part 
of both. Stating this principle with regard to propositions, it asserts 
that if a proposition implies each of two others, then it implies their 
joint assertion or logical product ; and this is the principle which was 
called composition above. 

35. From this point, we advance successfully until we require the 
idea of negation. This is taken, in the edition of the Formulaire we are 
considering, as a new primitive idea, and disjunction is defined by its 
means. By means of the negation of a proposition, it is of course easy 
to define the negation of a class : for " ^ is a not-a *" is equivalent to " oc 
is not an a.''"' But we require an axiom to the effect that not-a is a 
class, and another to the effect that not-not-a is a. Peano gives also a 
third axiom, namely : If a, h^ c be classes, and ah is contained in c, and oo 
is an a but not a c, then oc is not a h. This is simpler in the form : If ^, 
^, r be propositions, and p^ q together imply r, and q is true while r is 
false, then \ is false. This would be still further improved by being put 
in the form : If q^ r are propositions, and q implies r, then not-r implies 
not-g' ; a form which Peano obtains as a deduction. By dealing with 
propositions before classes or propositional functions, it is possible, as we 
saw, to avoid treating negation as a primitive idea, and to replace all 
axioms respecting negation by the principle of reduction. 

We come next to the definition of the disjunction or logical sum of 
two classes. On this subject Peano has many times changed his 
^ See e.g. F. 1901, Part I, § 1, Prop. 3. 3 (p. 10). 



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32 The Indefinahles of Mathematics [chap, ii 

procedure. In the edition we are considering, " a or h " is defined as the 
negation of the logical product of not-^ and not-^, i.e, as the class of 
terms which are not both woi-a and not-6. In later editions {e.g. F. 1901^ 
p. 19), we find a somewhat less artificial definition, namely : " a or b "' 
consists of all terms which belong to any class which contains a and 
contains b. Either definition seems logically unobjectionable. It is to 
be observed that a and b are classes, and that it remains a question for 
philosophical logic whether there is not a quite different notion of the 
disjunction of individuals, as ^.^. "Brown or Jones." I shall consider 
this question in Chapter v. It will be remembered that, when we begin 
by the calculus of propositions, disjunction is defined before negation ; 
with the above definition (that of 1897), it is plainly necessary to take 
negation first. 

36. The connected notions of the null-class and the existence of a 
class are next dealt with. In the edition of 1897, a class is defined as 
null when it is contained in every class. When we remember the 
definition of one class a being contained in another b (" a^ is an a'''' 
implies "^ is a 6'^ for all values of <r), we see that we are to regard 
the implication as holding for all values, and not only for those values 
for which ^ really is an a. This is a point upon which Peano is not 
explicit, and I doubt whether he has made up his mind on it. If the 
implication were only to hold when ^ really is an a^ it would not give >a 
definition of the null-class, for which this hypothesis is false for all values 
of a?. I do not know whether it is for this reason or for some other that 
Peano has since abandoned the definition of the inclusion of classes 
by means of formal implication between propositional functions : the 
inclusion of classes appears to be now regarded as indefinable. Another 
definition which Peano has sometimes favoured (e.g. F. 1895, Errata, 
p. 116) is, that the null-class is the product of any class into its 
negation — a definition to which similar remarks apply. In R. d. M. vii, 
No. 1 (§ 3, Prop. 1. 0), the null-class is defined as the class of those terms 
that belong to every class, i.e. the class of terms x such that "a is a 
class "' implies " ^ is an a " for all values of a. There are of course no 
such terms x ; and there is a grave logical difficulty in trying to interpret 
extensionally a class which has no extension. This point is one to which 
I shall return in Chapter vi. 

From this point onward, Peano's logic proceeds by a smooth develop- 
ment. But in one respect it is still defective : it does not recognize as 
ultimate relational propositions not asserting membership of a class. 
For this reason, the definitions of a function* and of other essentially 
relational notions are defective. But this defect is easily remedied by 
applying, in the manner explained above, the principles of the 
Formidaire to the logic of relations f. 

•5^ E.g. F. 1901, Part I, § 10, Props. 1. 0. 01 (p. 33). 

t See my article ^^ Sur la logique des relations/* R. d. M. Vol. vii, 2 (1901). 



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CHAPTER III. 

IMPLICATION AND FORMAL IMPLICATION. 

37. In the preceding chapter I endeavoured to present, briefly and 
uncritically, all the data, in the shape of formally fundamental ideas 
and propositions, that pure mathematics requires. In subsequent Parts 
I shall show that these • are all the data by giving definitions of the 
various mathematical concepts — number, infinity, continuity, the various 
spaces of geometry, and motion. In the remainder of Part I, I shall 
give indications, as best I can, of the philosophical problems arising in 
the analysis of the data, and of the directions in which I imagine these 
problems to be probably soluble. Some logical notions will be elicited 
which, though they seem quite fundamental to logic, are not commonly 
discussed in works on the subject ; and thus problems no longer clothed 
in mathematical symbolism will be presented for the consideration of 
philosophical logicians. 

Two kinds of implication, the material and the formal, were found to 
be essential to every kind of deduction. In the present chapter I wish 
to examinee and distinguish these two kinds, and to discuss some methods 
of attempting to analyze the second of them. 

In the discussion of inference, it is common to permit the intrusion 
of a psychological element, and to consider our acquisition of new 
knowledge by its means. But it is plain .that where we validly infer one 
proposition from another, we do so in virtue of a relation which holds 
between the two propositions whether we- perceive it or not : the mind, 
in fact, is as purely receptive in inference as common sense supposes it to 
be in perception of sensible objects. The relation in virtue of which it 
is possible for us validly to infer is what I call material implication. 
We have already seen that it would be a vicious circle to define this 
relation as meaning that %f one proposition is true, then another is true, 
for i/^ and then already involve implication. The relation holds, in fact, 
when it does hold, without any reference to the truth or falsehood of the 
propositions involved. 

But in developing the consequences of our assumptions as to impli- 
cation, we were led to conclusions which do not by any means agree with 

R. 3 



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34 The Indefinables of Mathematics [chap, hi 

what is commonly held concerning implication, for we found that any 
false proposition implies every proposition and any true proposition is 
implied by every proposition. Thus propositions are formally like a set 
of lengths each of which is one inch or two, and implication is like the 
relation "equal to or less than*" among such lengths. It would certainly 
not be commonly maintained that "2 + 2 = 4'*' can be deduced from 
"Socrates is a man," or that both are implied by "Socrates is a triangle.'"' 
But the reluctance to admit such implications is chiefly due, I think, to 
preoccupation with formal implication, which is a much more familiar 
notion, and is really before the mind, as a rule, even where material 
implication is what is explicitly mentioned. In inferences from "Socrates 
is a man,"*' it is customary not to consider the philosopher who vexed the 
Athenians, but to regard Socrates merely as a symbol, capable of being 
replaced by any other man ; and only a vulgar prejudice in favour of 
true propositions stands in the way of replacing Socrates by a number, a 
table, or a plum -pudding. Nevertheless, wherever, as in Euchd, one 
particular proposition is deduced from another, material implication is 
involved, though as a rule the material implication may be regarded as a 
particular instance of some formal implication, obtained by giving some 
constant value to the variable or variables involved in the said formal 
implication. And although, while relations are still regarded with the 
awe caused by unfamiliarity, it is natural to doubt whether any such 
relation as implication is to be found, yet, in virtue of the general- 
principles laid down in Section C of the preceding chapter, there must 
be a relation holding between nothing except propositions, and holding 
between any two propositions of which either the first is false or the 
second true. Of the various equivalent relations satisfying these 
conditions, one is to be called implication^ and if such a notion seems 
unfamiliar, that does not suffice to prove that it is illusory. 

38. At this point, it is necessary to consider a very difficult 
logical problem, namely, the distinction between a proposition actually 
asserted, and a proposition considered merely as a complex concept. 
One of our indemonstrable principles was, it will be remembered, that 
if the hypothesis in an implication is true, it may be dropped, and the 
consequent asserted. This principle, it was observed, eludes formal 
statement, and points to a certain failure of formalism in general. The 
principle is employed whenever a proposition is said to be proved ; for 
what happens is, in all such cases, that the proposition is shown to be 
implied by some true proposition. Another form in which the principle 
is constantly employed is the substitution of a constant, satisfying the 
hypothesis, in the consequent of a formal implication. If ^x implies y\rx 
for all values of ^, and if 6J^ is a constant satisfying ^x^ we can assert 
-^tt, dropping the true hypothesis (f>a. This occurs, for example, when- 
ever any of those rules of inference which employ the hypothesis 
that the variables involved are propositions, are applied to particular 



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37-39] Implication and Formal Implication 35 

propositions. The principle in question is, therefore, quite vital to any 
kind of demonstration. 

The independence of this principle is brought out by a consideration 
of Lewis Carroll's puzzle, "What the Tortoise said to Achilles*.'' The 
principles of inference which we accepted lead to the proposition that, if 
p and q be propositions, then p together with "p implies g^" implies q. 
At first sight, it might be thought that this would enable us to assert q 
provided p is true and implies q. But the puzzle in question shows that 
this is not the case, and that, until we have some new principle, we' shall 
only be led into an endless regress of more and more complicated impli- 
cations, without ever arriving at the assertion of q. We need, in fact, 
the notion of therefore^ which is quite different from the notion of implies^ 
and holds between different entities. In grammar, the distinction is that 
between a verb and a verbal noun, between, say, '' A is greater than B " 
and " J's being greater than B^ In the first of these, a proposition is 
actually asserted, whereas in the second it is merely considered. But 
these are psychological terms, whereas the difference which I desire to 
express is genuinely logical. It is plain that, if I may be allowed to 
use the word assertion in a non-psychological sense, the proposition 
" p implies q " asserts an implication, though it does not assert p or q. 
The p and the q which enter into this proposition are not strictly the 
same as the^ or the q which are separate propositions, at least, if they 
are true. The question is : How does a proposition differ by being 
actually true from what it would be as an entity if it were not true ? It 
is plain that true and false propositions alike are entities of a kind, but 
that true propositions have a quality not belonging to false ones, a. 
quality which, in a non-psychological sense, may be called being 
asserted. Yet there are grave difficulties in forming a consistent theory 
on this point, for if assertion in any way changed a proposition, no 
proposition which can possibly in any context be unasserted could be 
true, since when asserted it would become a different proposition. But 
this is plainly false ; for in " p implies g," p and q are not asserted, and 
yet they may be true. Leaving this puzzle to logic, however, we must 
insist that there is a difference of some kind between an asserted and an 
unasserted proposition f. When we say therefore^ we state a relation 
which can only hold between asserted propositions, and which thus 
differs from implication. Wherever therefore occurs, the hypothesis 
may be dropped, and the conclusion asserted by itself. This seems to 
be the first step in answering Lewis Carroll's puzzle. 

39. It is commonly said that an inference must have premisses 
and a conclusion, and it is held, apparently, that two or more premisses 
are necessary, if not to all inferences, yet to most. This view is borne 
out, at first sight, by obvious facts : every syllogism, for example, is held 

■^ Mind, N. S. Vol. iv, p. 278. 

t Frege {loc. cit.) has a special symbol to denote assertion. 

3—2 



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36 The Indejinahles of Mathematics [chap, hi 

to have two premisses. Now such a theory greatly complicates the 
relation of implication, since it renders it a relation which may have any 
number of terms, and is symmetrical with respect to all but one of them, 
but not symmetrical with respect to that one (the conclusion). This 
complication is, however, unnecessary, first, because every simultaneous 
assertion of a number of propositions is itself a single proposition, and 
secondly, because, by the rule which we called eocportation^ it is always 
possible to exhibit an implication explicitly as holding between single 
propositions. To take the first point first : if A; be a class of proposi- 
tions, all the propositions of the class h are asserted by the single 
proposition "for all values of oc^ if oc implies ^, then '^c is a A;' implies 
^5" or, in more ordinary language, " every h is true." And as regards 
the second point, which assumes the number of premisses to be finite, 
" pq implies r'" is equivalent, if g be a proposition, to " p implies that q 
implies r,'** in which latter form the implications hold explicitly between 
single propositions. Hence we may safely hold implication to be a 
relation between two propositions, not a relation of an arbitrary number 
of premisses to a single conclusion. 

40. I come now to formal implication, which is a far more difiicult 
notion than material implication. In order to avoid the general notion 
of propositional function, let us begin by the discussion of a particular 
instance, say "x is a man implies x is a mortal for all values of oc.'''' 
This proposition is equivalent to " all men are mortal " " every man is 
mortar*" and "any man is mortal.'' But it seems highly doubtful 
whether it is the same proposition. It is also connected with a purely 
intensional proposition in which man is asserted to be a complex notion 
of which mortal is a constituent, but this proposition is quite distinct 
from the one we are discussing. Indeed, such intensional propositions 
are not always present where one class is included in another : in general, 
either class may be defined by various different predicates, and it is by 
no means necessary that every predicate of the smaller class should 
contain every predicate of the larger class as a factor. Indeed, it may 
very well happen that both predicates are philosophically simple : thus 
colour and existent appear to be both simple, yet the class of colours is 
part of the class of existents. The intensional view, derived from 
predicates, is in the main irrelevant to Symbolic Logic and to Mathe- 
matics, and I shall not consider it further at present. 

41. It may be doubted, to begin with, whether "^ is a man 
implies ^ is a mortal '^ is to be regarded as asserted strictly of all possible 
terms, or only of such terms as are men. Peano, though he is not explicit, 
appears to hold the latter view. But in this case, the hypothesis ceases 
to be significant, and becomes a mere definition of ^ : x is to mean any 
man. The hypothesis then becomes a mere assertion concerning the 
meaning of the symbol x^ and the whole of what is asserted concerning 
the matter dealt with by our symbol is put into the conclusion. The 



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39-41] Implication and Formal Implication 37 

premiss says : ^ is to mean any man. The conclusion says : x is mortaL 
But the implication is merely concerning the symbolism : since any man 
is mortal, if oc denotes any man, oo is mortal. Thus formal implication^ 
on this view, l\p.s wholly disappeared, leaving us the proposition " any 
man is mortal" as expressing the whole of what is relevant in the 
proposition with a variable. It would now only remain to examine 
the proposition " any man is mortal," and if possible to explain this 
proposition without reintroducing the variable and formal implication. 
It must be confessed that some grave difficulties are avoided by this 
view. Consider, for example, the simultaneous assertion of all the 
propositions of some class h: this is not expressed by '"oc is a A;"* implies 
oc for all values of zf ." For as it stands, this proposition does not express 
what is meant, since, if x be not a proposition, " x is a ^' " cannot imply 
X ; hence the range of variability of x must be confined to propositions, 
unless we prefix (as above, § 39) the hypothesis " x implies .r." This 
remark applies generally, throughout the propositional calculus, to all 
cases where the conclusion is represented by a single letter : unless the 
letter does actually represent a proposition, the implication asserted will 
be false, since only propositions can be implied. The point is that, if x 
be our variable, x itself is a proposition for all values of x which are 
propositions, but not for other values. This makes it plain what the 
limitations are to which our variable is subject: it must vary only within 
the range of values for which the two sides of the principal implication 
are propositions, in other words, the two sides, when the variable is not 
replaced by a constant, must be genuine propositional functions. If this 
restriction is not observed, fallacies quickly begin to appear. It should be 
noticed that there may be any number of subordinate implications which 
do not require that their terms should be propositions : it is only of the 
principal implication that this is required. Take, for example, the first 
principle of inference : If p implies g, then p implies q. This holds 
equally whether p and q be propositions or not ; for if either is not a 
proposition, " p implies q " becomes false, but does not cease to be a 
proposition. In fact, in virtue of the definition of a proposition, our 
principle states that " p implies g " is a propositional function, i.e. that 
it is a proposition for all values of p and q. But if we apply the 
principle of importation to this proposition, so as to obtain '" p implies 
9," together with j9, implies g," we have a formula which is only true 
when p and q are propositions : in order to make it true universally, we 
must preface it by the hypothesis " p implies p and q implies §'." In this 
way, in many cases, if not in all, the restriction on the variability of the 
variable can be removed ; thus, in the assertion of the logical product of 
a class of propositions, the formula "if x implies x^ then ""x is a k'' 
implies <r" appears unobjectionable, and allows x to vary without restric- 
tion. Here the subordinate implications in the premiss and the conclusion 
are material : only the principal implication is formal. 



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38 The Indefinables of Mathematics [chap, m 

Returning now to " ^ is a man implies z?7 is a mortal,'" it is plain that 
no restriction is required in order to insure our having a genuine pro- 
position. And it is plain that, although we might restrict the values of 
00 to men, and although this seems to be done in the proposition 
" all men are mortal,'** yet there is no reason, so far as the truth of our 
proposition is concerned, why we should so restrict our oc. Whether x 
be a man or not, " ^ is a man '' is always, when a constant is substituted 
for ^, a proposition implying, for that value of ^, the proposition " x is 
a mortal.'' And unless we admit the hypothesis equally in the cases 
where it is false, we shall find it impossible to deal satisfactorily with the 
null-class or with null propositional functions. We must, therefore, 
allow our >r, wherever the truth of our formal implication is thereby 
unimpaired, to take all values without exception; and where any 
restriction on variability is required, the implication is not to be 
regarded as formal until the said restriction has been removed by being 
prefixed as hypothesis. (If -^x be a proposition whenever x satisfies <^x^ 
where <^x is a propositional function, and if y\rx^ whenever it is a pro- 
position, implies ^^', then "y^rx implies %^" is not a formal implication, 
but " (^x implies that y^x implies %^" is a formal implication.) 

42. It is to be observed that "x is a man implies z27 is a mortal " 
is not a relation of two propositional functions, but is itself a single 
propositional function having the elegant property of being always 
true. For '' x is a man" is, as it stands, not a proposition at all, 
and does not imply anything!; and we must not first vary our x in 
"x is a man," and then independently vary it in "x is a mortal," 
for this would lead to the proposition that "everything is a man" 
implies >' everything is a. mortal," which, though true, is not what was 
meant. This proposition would have to be expressed, if the language 
of variables were retained, by two variables, as "^ is a man implies 
y is a mortal." But this formula too is unsatisfactory, for its natural 
meaning would be : "If anything is a man, then everything is a mortal." 
The point to be emphasized is, of course, that our x^ though variable, 
must be the same on both sides of the implication, and this requires 
that we should not obtain our formal implication by first varying (say) 
Socrates in "Socrates is a man," and then in "Socrates is a mortal," 
but that we should start from the whole proposition "Socrates is a 
man implies Socrates is a mortal," and vary Socrates in this proposition 
as a whole. Thus our formal implication asserts a class of implications, 
not a single implication at all. We do not, in a word, have one im- 
plication containing a variable, but rather a variable implication. We 
have a class of implications, no one of which contains a variable, and 
we assert that every member of this class is true. This is a first step 
towards the analysis of the mathematical notion of the variable. 

But, it may be asked, how comes it that Socrates may be varied 
in the proposition " Socrates is a man implies Socrates is mortal".'^ In 



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41-44] Implication and Formal Iirvplication 39 

virtue of the fact that true propositions are implied by all others, we 
have "Socrates is a man implies Socrates is a philosopher"; but in this 
proposition, alas, the variability of Socrates is sadly restricted. This 
seems to show that formal implication involves something over and 
above the relation of implication, and that some additional relation 
must hold where a term can be varied. In the case in question, it is 
natural to say that what is involved is the relation of inclusion between 
the classes men and mortals — the very relation which was to be defined 
and explained by our formal implication. But this view is too simple 
to meet all cases, and is therefore not required in any case. A larger 
number of cases, though still not all cases, can be dealt with by the 
notion of what I shall call assertions. This notion must now be briefly 
explained, leaving its critical discussion to Chapter vii. 

43. It has always been customary to divide propositions into 
subject and predicate; but this division has the defect of omitting the 
verb. It is true that a graceful concession is sometimes made by loose 
talk about the copula, but the verb deserves far more respect than is 
thus paid to it. We may say, broadly, that every proposition may be 
divided, some in only one way, some in several ways, into a term (the 
subject) arid something which is said about the subject, which something 
I shall call the assertion. Thus " Socrates is a man "'' may be divided 
into Socrates and is a man. The verb, which is the distinguishing mark 
of propositions, remains with the assertion; but the assertion itself, 
being robbed of its subject, is neither true nor false. In logical dis- 
cussions, the notion of assertion often occurs, but as the word proposition 
is used for it, it does not obtain separate consideration. Consider, for 
example, the best statement of the identity of indiscernibles : " If ^ and y 
be any two diverse entities, some assertion holds of x which does not 
hold of 2/." But for the word assertion^ which would ordinarily be 
replaced by proposition^ this statement is one which would commonly 
pass unchallenged. Again, it might be said: "Socrates was a philo- 
sopher, and the same is true of Plato."" Such statements require the 
analysis of a proposition into an assertion and a subject, in order that 
there may be something identical which can be said to be affirmed of 
two subjects. 

44. We can now see how, where the analysis into subject and 
assertion is legitimate, to distinguish implications in which there is a 
term which can be varied from others in which this is not the case. Two 
ways of making the distinction may be suggested, and we shall have to 
decide between them. It may be said that there is a relation between 
the two assertions "is a man'' and "is a mortal,'' in virtue of which, 
when the one holds, so does the other. Or again, we may analyze the 
whole proposition "Socrates is a man implies Socrates is a mortal " into 
Socrates and an assertion about him, and say that the assertion in 
question holds of all terms. Neither of these theories replaces the above 



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40 The Indefinahles of Mathematics [chap, hi 

analysis of "^ is a man implies ^ is a mortal " into a class of material 
implications ; but whichever of the two is true carries the analysis one 
step further. The first theory suffers from the difficulty that it is 
essential to the relation of assertions involved that both assertions 
should be made of the same subject, though it is otherwise irrelevant 
what subject we choose. The second theory appears objectionable on 
the ground that the suggested analysis of " Socrates is a man implies 
Socrates is a mortaP' seems scarcely possible. The proposition in 
question consists of two terms and a relation, the terms being "Socrates 
is a man" and "Socrates is a mortal"; and it would seem that when a 
relational proposition is analyzed into a subject and an assertion, the 
subject must be one of the terms of the relation which is asserted. This 
objection seems graver than that against the former view; I shall 
therefore, at any rate for the present, adopt the former view, and regard 
formal implication as derived from a relation between assertions. 

We remarked above that the relation of inclusion between classes is 
insufficient. This results from the irreducible nature of relational 
propositions. Take e.g. "Socrates is married implies Socrates had a 
father." Here it is affirmed that because Socrates has one relation, 
he must have another. Or better still, take "A is before B implies B is 
after J." This is a formal implication, in which the assertions are 
(superficially at least) concerning different subjects; the only way to 
avoid this is to say that both propositions have both A and B as 
subjects, which, by the way, is quite different from saying that they 
have the one subject "A and jB." Such instances make it plain that 
the notion of a propositional function, and the notion of an assertion, 
are more fundamental than the notion of cZ<2^^,. and that the latter is 
not adequate to explain all cases of formal implication. I shall not 
enlarge upon this point now, as it will be abundantly illustrated in 
subsequent portions of the present work. 

It is important to realize that, according to the above analysis of 
formal implication, the notion of every term is indefinable and ultimate. 
A formal implication is one which holds of every term, and therefore 
every cannot be explained by means of formal implication. If a and h 
be classes, we can explain "every a is a 6" by means of "^ is an a 
implies x is a 6"; but the every which occurs here is a derivative and 
subsequent notion, presupposing the notion of every term. It seems 
to be the very essence of what may be called a formal truth, and of 
formal reasoning generally, that some assertion is affirmed to hold of 
every term; and unless the notion oi every term is admitted, formal 
truths are impossible. 

45. The fundamental importance of formal implication is brought 
out by the consideration that it is involved in all the rules of inference. 
This shows that we cannot hope wholly to define it in terms of material 
implication, but that some further element or elements must be involved. 



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44, 45] Implication and Formal Implication 41 

We may observe, however, that, in a particular inference, the rule 
according to which the inference proceeds is not required as a premiss. 
This point has been emphasized by Mr Bradley*; it is closely connected 
with the principle of dropping a true premiss, being again a respect 
in which formalism breaks down. In order to apply a rule of inference, 
it IS formally necessary to have a premiss asserting that the present 
case is an instance of the rule; we shall then need to affirm the rule by 
which we can go from the rule to an instance, and also to affirm that here 
we have an instance of this rule, and so on into an endless process. 
The fact is, of course, that any implication warranted by a rule of 
inference does actually hold, and is not merely implied by the rule. 
This is simply an instance of the non-formal principle of dropping a 
true premiss: if our rule implies a certain implication, the rule may be 
dropped and the implication asserted. But it remains the case that the 
fact that our rule does imply the said implication, if introduced at all, 
must be simply perceived, and is not guaranteed by any formal deduction ; 
and often it is just as easy, and consequently just as legitimate, to perceive 
immediately the implication in question as to perceive that it is implied 
by one or more of the rules of inference. 

To sum up our discussion of formal implication : a formal implication, 
we said, is the affirmation of every material implication of a certain 
class; and the class of material implications involved is, in simple cases, 
the class of all propositions in which a given fixed assertion, made con- 
cerning a certain subject or subjects, is affirmed to imply another given 
fixed assertion concerning the same subject or subjects. Where a formal 
implication holds, we agreed to regard it, wherever possible, as due to 
some relation between the assertions concerned. This theory raises many 
formidable logical problems, and requires, for its defence, a thorough 
analysis of the constituents of propositions. To this task we must now 
address ourselves. 

^ Logic, Book II, Part I, Chap, ii (p. 227). 



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CHAPTEE IV. 



PROPER NAMES, ADJECTIVES, AND VERBS. 

46. In the present chapter, certain questions are to be discussed 
belonging to what may be called philosophical grammar. The study 
of grammar, in my opinion, is capable of throwing far more light on 
philosophical questions than is commonly supposed by philosophers. 
Although a grammatical distinction cannot be uncritically assumed to 
correspond to a genuine philosophical difference, yet the one is prima 
facie evidence of the other, and may often be most usefully employed 
as a source of discovery. Moreover, it must be admitted, I think, that 
every word occurring in a sentence must have some meaning : a perfectly 
meaningless sound could not be employed in the more or less fixed 
way in which language employs words. The correctness of our philo- 
sophical analysis of a proposition may therefore be usefully checked 
by the exercise of assigning the meaning of each word in the sentence 
expressing the proposition. On the whole, grammar seems to me to 
bring us much nearer to a correct logic than the current opinions of 
philosophers; and in what follows, grammar, though not our master, 
will yet be taken as our guide*. 

Of the parts of speech, three are specially important : substantives, 
adjectives, and verbs. Among substantives, some are derived from 
adjectives or verbs, as humanity from human, or sequence from follows, 
(I am not speaking of an etymological derivation, but of a logical one.) 
Others, such as proper names, or space, time, and matter, are not 
derivative, but appear primarily as substantives. What we wish to 
obtain is a classification, not of words, but of ideas ; I shall therefore 
call adjectives or predicates all notions which are capable of being such, 
even in a form in which grammar would call them substantives. The 
fact is, as we shall see, that human and humanity denote precisely 
the same concept, these words being employed respectively according to 
the kind of relation in which this concept stands to the other constituents 
of a proposition in which it occurs. The distinction which we require 

"^ Tlie excellence of grammar as a gui4e is proportional to the paucity of 
inflexions^ i.e. to the degree of analysis effected by the language considered. 



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46, 47] Proper Names, Adjectives, and Verbs 43 

is not identical with the grammatical distinction between substantive 
and adjective, since one single concept may, according to circumstances, 
be either substantive or adjective : it is the distinction between proper 
and general names that we require, or rather between the objects in- 
dicated by such names. In every proposition, as we saw in Chapter iii, 
we may make an analysis into something asserted and something about 
which the assertion is made. A proper name, when it occurs in a 
proposition, is always, at least according to one of the possible ways 
of analysis (where there are several), the subject that the proposition 
or some subordinate constituent proposition is about, and not what is 
said about the subject. Adjectives and verbs, on the other hand, 
are capable of occurring in propositions in which they cannot be 
regarded as subject, but only as parts of the assertion. Adjectives 
are distinguished by capacity for denoting — a term which I intend 
to use in a technical sense to be discussed in Chapter v. Verbs 
are distinguished by a special kind of connection, exceedingly hard 
to define, with truth and falsehood, in virtue of which they dis- 
tinguish an asserted proposition from an unasserted one, e.g. " Caesar 
died '' from " the death of Caesar.'' These distinctions must now be 
amplified, and I shall begin with the distinction between general and 
proper names. 

47. Philosophy is familiar with a certain set of distinctions, all 
more or less equivalent : I mean, the distinctions of subject and pre- 
dicate, substance and attribute, substantive and adjective, this and 
what^. I wish now to point out briefly what appears to me to be the 
truth concerning these cognate distinctions. The subject is important, 
since the issues between monism and monadism, between idealism and 
empiricism, and between those who maintain and those who deny that 
all truth is concerned with what exists, all depend, in whole or in part, 
upon the theory we adopt in regard to the present question. But the 
subject is treated here only because it is essential to any doctrine of 
number or of the nature of the variable. Its bearings on general 
philosophy, important as they are, will be left wholly out of account. 

Whatever may be an object of thought, or may occur in any true 
or false proposition, or can be counted as one, I call a term. This, 
then, is the widest word in the philosophical vocabulary. I shall use 
as synonymous with it the words unit, individual, and entity. The 
first two emphasize the fact that every term is one, while the third is 
derived from the fact that every term has being, i.e. is in some sense. 
A man, a moment, a number, a class, a relation, a chimaera, or anything 
else that can be mentioned, is sure to be a term ; and to deny that such 
and such a thing is a term must always be false. 

It might perhaps be thought that a word of such extreme generality 
could not be of any great use. Such a view, however, owing to certain 
* This last pair of terms is due to Mr Bradley. 



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44 The Indejinables of Mathematics [chap, iv 

wide-spread philosophical doctrines, would be erroneous. A term is, 
in fact, possessed of all the properties commonly assigned to substances 
or substantives. Every term, to begin with, is a logical subject : it is^ 
for example, the subject of the proposition that itself is one. Again 
every term is inmiutable and indestructible. What a term is, it is, and 
no change can be conceived in it which would not destroy its identity 
and make it another term*. Another mark which belongs to terms 
is numerical identity with themselves and numerical diversity from all 
other terms f. Numerical identity and diversity are the source of unity 
and plurality ; and thus the admission of many terms destroys monism. 
And it seems undeniable that every constituent of every proposition can 
be counted as one, and that no proposition contains less than two 
constituents. Term is, therefore, a useful word, since it marks dissent 
from various philosophies, as well as because, in many statements, we 
wish to speak of ajiy term or some term. 

48. Among terms, it is possible to distinguish two kinds, which 
I shall call respectively things and concepts. The former are the terms 
indicated by proper names, the latter those indicated by all other words. 
Here proper names are to be understood in a somewhat wider sense than 
is usual, and things also are to be understood as embracing all par- 
ticular points and instants, and many other entities not commonly called 
things. Among concepts, again, two kinds at least must be distinguished,, 
namely those indicated by adjectives and those indicated by verbs. The 
former kind will often be called predicates or class-concepts ; the latter 
are always or almost always relations. (In intransitive verbs, the notion 
expressed by the verb is complex, and usually asserts a definite relation 
to an indefinite relatura, as in ''Smith breathes.'') 

In a large class of propositions, we agreed, it is possible, in one or 
more ways, to distinguish a subject and an assertion about the subject. 
The assertion must always contain a verb, but except in this respect, 
assertions appear to have no universal properties. In a relational 
proposition, say "A is greater than jB," we may regard A as the subject,, 
and " is greater than B'''' as the assertion, or B as the subject and "^ is 
greater than" as the assertion. There are thus, in the case proposed^ 
two ways of analyzing the proposition into subject and assertion. 
Where a relation has more than two terms, as in " A is here now;]:,''^ 
there will be more than two ways of making the analysis. But in 
some propositions, there is only a single way: these are the subject- 

* The notion of a term here set forth is a modification of Mr G. E. Moore's, 
notion of a concept in his article ^'^On the Nature of Judgment/' Mind, N. S. No. 30^ 
from which notion^ however^ it diifers in some important respects. 

t On identity^ see Mr G. E. Moore's article in the Proceedings of the Aristotelian. 
Society, 1900-1901. 

X This proposition means ^^ A is in this place at this time." It will he shown in 
Part VII that the relation expressed is not reducible to a two-term relation. 



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47-49] Proper Names, Adjectives, and Verbs 45 

predicate propositions, such as "Socrates is human."' The proposition 
" humanity belongs to Socrates," which is equivalent to '' Socrates is 
human,'' is an assertion about humanity; but it is a distinct propo- 
sition. In /'Socrates is human," the notion expressed by human occurs 
in a different way from that in which it occurs when it is called 
humanity y the difference being that in the latter case, but not in the 
former, . the proposition is about this notion. This indicates that 
humanity is a concept, not a thing. I shall speak of the terms of a 
proposition as those terms, however numerous, which occur in a propo- 
sition and may be regarded as subjects about which the proposition is. 
It is a characteristic of the terms of a proposition that any one of 
them may be replaced by any other entity without our ceasing to have 
a proposition. Thus we shall say that "Socrates is human" is a 
proposition having only one term ; of the remaining components of 
the proposition, one is the verb, the other Is a predicate. With the sense 
which is has in this proposition, we no longer have a proposition at all 
if we replace human by something other than a predicate. Predicates, 
then, are concepts, other than verbs, which occur in propositions having 
only one term or subject. Socrates is a thing, because Socrates can 
never occur otherwise than as term in a proposition : Socrates is not 
capable of that curious twofold use which is involved in human and 
humanity. Points, instants, bits of matter, particular states of mind^ 
and particular existents generally, are things in the above sense, and 
so are many terms which do not exist, for example, the points in a 
non-Euclidean space and the pseudo-existents of a novel. All classes, 
it would seem, as numbers, men, spaces, etc., when taken as single terms, 
are things ; but this is a point for Chapter vi. 

Predicates are distinguished from other terms by a number of very 
interesting properties, chief among which is their connection with what 
I shall call denoting. One predicate always gives rise to a host of 
cognate notions : thus in addition to human and humanity, which 
only differ grammatically, we have m.an, a man, some man, any man, 
every man, all men^, all of which appear to be genuinely distinct one 
from another. The study of these various notions is absolutely vital 
to any philosophy of mathematics ; and it is on account of them that 
the theory of predicates is important. 

49. It might be thought that a distinction ought to be made 
between a concept as such and a concept used as a term, between, 
e.g., such pairs as is and being, human and humanity, one in such a 
proposition as " this is one " and 1 in " 1 is a number." But inex- 
tricable difficulties will envelop us if we allow such a view. There is, 

* I use all men as collective^ i.e. as nearly synonymous with the human race, but 
differing therefrom by being many and not one. I shall always use all collectively^ 
confining myself to every for the distributive sense. Thus I shall say ^'^ every man is 
mortal_," not ^^all men are mortal." 



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46 The Indefinables of Mathematics [chap, iy 

of course, a grammatical difference, and this corresponds to a difference 
as regards relations. In the first case, the concept in question is used 
as a concept, that is, it is actually predicated of a term or asserted to 
relate two or more terms ; while in the second case, the concept is 
itself said to have a predicate or a relation. There is, therefore, 
no difficulty in accounting for the grammatical difference. But what 
I wish to urge is, that the difference lies solely in external relations, 
and not in the intrinsic nature of the terms. For suppose that one 
as adjective differed from 1 as term. In this statement, one as 
adjective has been made into a term ; hence either it has become 
1, in which case the supposition is self-contradictory; or there is some 
other difference between one and 1 in addition to the fact that the 
first denotes a concept not a term while the second denotes a concept 
which is a term. But in this latter hypothesis, there must be propo- 
sitions concerning one as term, and we shall still have to maintain 
propositions concerning one as adjective as opposed to one as term ; 
yet all such propositions must be false, since a proposition about one 
as adjective makes one the subject, and is therefore really about one 
as term. In short, if there were any adjectives which could not be 
made into substantives without change of meaning, all propositions 
concerning such adjectives (since they would necessarily turn them into 
substantives) would be false, and so would the proposition that all 
such propositions are false, since this itself turns the adjectives into 
substantives. But this state of things is self-contradictory. 

The above argument proves that we were right in saying that terms 
embrace everything that can occur in a proposition, with the possible 
exception of complexes of terms of the kind denoted by any and cognate 
words*. For if A occurs in a proposition, then, in this statement, 
A is the subject; and we have just seen that, if A is ever not the 
subject, it is exactly and numerically the same A which is not subject 
in one proposition and is subject in another. Thus the theory that 
there are adjectives or attributes or ideal things, or whatever they may 
be called, which are in some way less substantial, less self-subsistent, 
less self-identical, than true substantives, appears to be wholly erroneous, 
and to be easily reduced to a contradiction. Terms which are concepts 
differ from those which are not, not in respect of self-subsistence, but 
in virtue of the fact that, in certain true or false propositions, they 
occur in a manner which is different in an indefinable way from the 
manner in which subjects or terms of relations occur. 

50. Two concepts have, in addition to the numerical diversity 
which belongs to them as terms, another special kind of diversity 
which may be called conceptual. This may be characterized by the 
fact that two propositions in which the concepts occur otherwise than 
as terms, even if, in all other respects, the two propositions are identical, 

"^ See the next chapter. 



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49-52] Froper Names, AdjectiveSy and Verbs 47 

yet differ in virtue of the fact that the concepts which occur in 
them are conceptually diverse. Conceptual diversity implies numerical 
diversity, but the converse implication does not hold, since not all 
terms are concepts. Numerical diversity, as its name implies, is the 
source of plurality, and conceptual diversity is less important to 
mathematics. But the whole possibility of making different assertions 
about a given term or set of terms depends upon conceptual diversity, 
which is therefore fundamental in general logic. 

51. It is interesting and not unimportant to examine very briefly 
the connection of the above doctrine of adjectives with certain traditional 
views on the nature of propositions. It is customary to regard all 
propositions as having a subject and a predicate, i.e. as having an 
immediate this, and a general concept attached to it by way of description. 
This is, of course, an account of the theory in question which will strike 
its adherents as extremely crude; but it will serve for a general indication 
of the view to be discussed. This doctrine develops by internal logical 
necessity into the theory of Mr Bradley's Logic, that all words stand for 
ideas having what he calls meaning, SiwA that in every judgment there 
is a something, the true subject of the judgment, which is not an idea 
and does not have meaning. To have meaning, it seems to me, is a 
notion confusedly compounded of logical and psychological elements. 
Words all have meaning, in the simple sense that they are symbols 
which stand for something other than themselves. But a proposition, 
unless it happens to be linguistic, does not itself contain words : it 
contains the entities indicated by words. Thus meaning, in the sense 
in which words have meaning, is irrelevant to logic. But such concepts 
as a man have meaning in another sense : they are, so to speak, symbolic 
in their own logical nature, because they have the property which I call 
denoting. That is to say, when a man occurs in a proposition {e.g, 
" I met a man in the street ""), the proposition is not about the concept 
a man, but about something quite different, some actual biped denoted 
by the concept. Thus concepts of this kind have meaning in a non- 
psychological sense. And in this sense, when we say " this is a man,*" 
we are making a proposition in which a concept is in some sense 
attached to what is not a concept. But when meaning is thus under- 
stood, the entity indicated by John does not have meaning, as Mr Bradley 
contends * ; and even among concepts, it is only those that denote that 
have meaning. The confusion is largely due, I believe, to the notion 
that words occur in propositions, which in turn is due to the notion that 
propositions are essentially mental and are to be identified with cognitions. 
But these topics of general philosophy must be pursued no further in 
this work. 

52. It remains to discuss the verb, and to find marks by which 
it is distinguished from the adjective. In regard to verbs also, there is 

* Logic, Book I, Chap, i, §§ 17, 18 (pp. 58-60). 



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48 The Indefinables of Mathematics [chap. IV 

a twofold grammatical form corresponding to a difference in merely 
external relations. There is the verb in the form which it has as verb 
(the various inflexions of this form may be left out of account), and 
there is the verbal noun, indicated by the infinitive or (in English) the 
present participle. The distinction is that between "Felton killed 
Buckingham '' and " Killing no murder." By analyzing this difference, 
the nature and function of the verb will appear. 

It is plain, to begin with, that the concept which occurs in the verbal 
noun is the very same as that which occurs as verb. This results from 
the previous argument, that every constituent of every proposition must, 
on pain of self-contradiction, be capable of being made a logical subject. 
If we say '' hills does not mean the same as to hill^'' we have already 
made Mils a subject, and we cannot say that the concept expressed by 
the word Mils cannot be made a subject. Thus the very verb which 
occurs as verb can occur also as subject. The question is : What logical 
difference is expressed by the difference of grammatical form ? And it 
is plain that the difference must be one in external relations. But 
in regard to verbs, there is a further point. By transforming the verb, 
as it occurs in a proposition, into a verbal noun, the whole proposition 
can be turned into a single logical subject, no longer asserted, and no 
longer containing in itself truth or falsehood. But here too, there seems 
to be no possibility of maintaining that the logical subject which results 
is a different entity from the proposition. " Caesar died " and " the 
death of Caesar " will illustrate this point. If we ask : What is asserted 
in the propo'sition " Caesar died " ? the answer must be " the death of 
Caesar is asserted.'^ In that case, it would seem, it is the death of Caesar 
which is true or false; and yet neither truth nor falsity belongs to 
a mere logical subject. The answer here seems to be that the death of 
Caesar has an external relation to truth or falsehood (as the case may 
be), whereas " Caesar died "" in some way or other s^ontains its own truth 
or falsehood as an element. But if this is the correct analysis, it is 
difficult to see how " Caesar died'' differs from "the truth of Caesar's 
death " in the case where it is true, or " the falsehood of Caesar's death " 
in the other case. Yet it is quite plain that the latter, at any rate, is 
never equivalent to " Caesar died." There appears to be an ultimate 
notion of assertion, given by the verb, which is lost as soon as we 
substitute a verbal noun, and is lost when the proposition in question 
is made the subject of some other proposition. This does not depend 
upon grammatical form ; for if I say " Caesar died is a proposition," 
I do not assert that Caesar did die, and an element which is present in 
" Caesar died " has disappeared. Thus the contradiction which was to 
have been avoided, of an entity which cannot be made a logical subject, 
appears to have here become inevitable. This difficulty, which seems to 
be inherent in the very nature of truth and falsehood, is one with which 
I do not know how to deal satisfactorily. The most obvious course 



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52-54] Proper Names, Adjectives, and Verbs 49 

would be to say that the difference between an asserted and an unasserted 
proposition is not logical, but psychological. In the sense in which 
false propositions may be asserted, this is doubtless true. But there 
is another sense of assertion, very difficult to bring clearly before the 
mind, and yet quite undeniable, in which only true propositions are 
asserted. True and false propositions alike are in some sense entities,, 
and are in some sense capable of being logical subjects; but when 
a proposition happens to be true, it has a further quality, over and 
above that which it shares with false propositions, arid it is this further 
quality which is what I mean by assertion in a logical as opposed to 
a psychological sense. The nature of truth, however, belongs no more 
to the principles of mathematics than to the principles of everything 
else. I therefore leave this question to the logicians with the above 
brief indication of a difficulty. 

53. It may be asked whether everything that, in the logical sense 
we are concerned with, is a verb, expresses a relation or not. It seems 
plain that, if we were right in holding that '^Socrates is human " is a 
proposition having only one term, the is in this proposition cannot 
express a relation in the ordinary sense. In fact, subject-predicate 
propositions are distinguished by just this non-relational character. 
Nevertheless, a relation between Socrates and humanity is certainly 
implied, and it is very difficult to conceive the proposition as expressing 
no relation at all. We may perhaps say that it is a relation, although 
it is distinguished from other relations in that it does not permit itself 
to be regarded as an assertion concerning either of its terms indifflsrently, 
but only as an assertion concerning the referent. A similar remark may 
apply to the proposition " A is,"" which holds of every term without 
exception. The is here is quite different from the is in " Socrates is 
human '''' ; it may be regarded as complex, and as really predicating 
Being oi A. In this ^#iy, the true logical verb in a proposition may be 
always regarded as asserting a relation. But it is so hard to .know 
exactly what is meant by relation that the whole question is in danger 
of becoming purely verbal. 

54. The twofold nature of the verb, as actual verb and as verbal 
noun, may be expressed, if all verbs are held to be relations, as the 
difference between a relation in itself and a relation actually relating. 
Consider, for example, the proposition "A differs from B^ The 
constituents of this proposition, if we analyze it, appear to be only A, 
difference^ J5. Yet these constituients, thus placed side by side, do not 
reconstitute the proposition. The difference which occurs in the 
proposition actually relates A and B, whereas the difference after 
analysis is a 'Motion which has no connection with A and B, It may 
be said that we ought, in the analysis, to mention the relations which 
difference has to A and B, relations which are expressed by is ecnA from 
when we say ''A is different from 5.*" These relations consist in the 

R. 4 



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50 T'he Indejinables of Mathematics [chap, iy 

fact that A is referent and B relatum with respect to difference. But 
"^5 referent, difference, relatum, 5" is still merely a list of terms, not 
a proposition. A proposition, in fact, is essentially a unity, and when 
analysis has destroyed the unity, no enumeration of constituents will 
restore the proposition. The verb, when used as a verb, embodies the 
unity of the proposition, and is thus distinguishable from the verb con- 
sidered as a term, though I do not know how to give a clear account of 
the precise nature of the distinction. 

55. It may be doubted whether the general concept difference 
Dccurs at all in the proposition "A differs from B^'' or whether there is 
not rather a specific diffference of A and B^ and another specific difference 
of C and D, which are respectively affirmed in "^ differs from 5'' and 
" C differs from D^ In this way, difference becomes a class-concept of 
which there are as many instances as there are pairs of different terms ; 
and the instances may be said, in Platonic phrase, to partake of the 
nature of difference. As this point is quite vital in the theory* of 
relations, it may be well to dwell upon it. And first of all, I must 
point out that in "A differs from j5" I intend to consider the bare 
numerical difference in virtue of which they are two, not difference in 
this or that respect. 

J^ Let us first try the hypothesis that a difference is a complex notion, 
compounded of difference together with some special quality distinguishing 
a particular difference from every other particular difference. So far as 
the relation of difference itself is concerned, we are to suppose that 
no distinction can be made between different cases ; but there are to be 
different associated qualities in different cases. But since cases are 
distinguished by their terms, the quality must be primarily associated 
with the terms, not with difference. If the quality be not a relation, it 
can have no special connection with the difference of A and 5, which it 
was to render distinguishable from bare difference, and if it fails in this 
it becomes irrelevant. On the other hand, if it be a new relation 
between A and 5, over and above difference, we shall have to hold that 
any two terms have two relations, difference and a specific difference, the 
latter not holding between any other pair of terms. This view is a 
combination of twp others, of which the first holds that the abstract 
general relation of difference itself holds between A and 5, while the 
second holds that when two terms differ they have, corresponding to 
this fact, a specific relation of difference, unique and unanalyzable and 
not shared by any other pair of terms. Either of these views may be 
held with either the denial or the affirmation of the other. Let us see 
what is to be said for arid against them. 

Against the notion of specific differences, it may be urged that, if 
differences differ, their differences from each other must also differ, and 
thus we are led into an endless process. Those who object to endless 
processes will see in this a proof that differences do not differ. But in 



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54, 55] Proper Names, Adjectives, and Verbs 51 

the present work, it will be maintained that there are no contradictions 
peculiar to the notion of infinity, and that an endless process is not to 
be objected to unless it arises in the analysis of the actual meaning of a 
proposition. In the present case, the process is one of implications, not 
one of analysis ; it must therefore be regarded as harmless. 

Against the notion that the abstract relation of difference holds 
between A and B, we have the argument derived from the analysis of 
" A differs from B^'' which gave rise to the present discussion. It is to 
be observed that the hypothesis which combines the general and the 
specific difference must suppose that there are two distinct propositions, 
the one affirming the general, the other the specific difference. Thus if 
there cannot be a general difference between A and B, this mediating 
hypothesis is also impossible. And we saw that the attempt to avoid 
the failure of analysis by including in the meaning of "A differs from ^" 
the relations of difference to A and B was vain. This attempt, in fact, 
leads tq an endless process of the inadmissible kind ; for we shall have to 
include the relations of the said relations to A and B and difference, and 
so on, and in this continually increasing complexity we are supposed 
to be only analyzing the meaning of our original proposition. This 
argument establishes a point of very great importance, namely, that 
when a relation holds between two terms, the relations of the relation to 
the terms, and of these relations to the relation and the terms, and so 
on ad infinitum^ though all impfied by the proposition affirming the 
original relation, form no part of the meaning of this proposition. 

But the above argument does not suffice to prove that the relation 
of A to B cannot be abstract difference : it remains tenable that, as 
was suggested to begin with, the true solution lies in regarding every 
proposition as having a kind of unity which analysis cannot preserve, 
and which is lost even though it be mentioned by analysis as an element 
in the proposition. This view has doubtless its own difficulties, but thf 
view that no two pairs of terms can have the same relation both contains 
difficulties of its own and fails to solve the difficulty for the sake of which 
it was invented. For, even if the difference of A and B be absolutely 
peculiar to A and^, still the three terms A, B, difference of A from B, 
do not reconstitute the proposition "A differs from 5," any more than 
A and B and difference did. And it seems plain that, even if differences 
did differ, they would still have to have something in common. But 
the most general way in which two terms can have something in common 
is by both having a given relation to a given term. Hence if no two 
pairs of terms can have the same relation, it follows that no two terms 
can have anything in common, and hence different differences will not 
be in any definable sense instances of difference *. I conclude, then, that 

* The above argument appears to prove that Mr Moore's theory of universals 
with numerically diverse instances in his paper on Identity {Pi^oceedings of the 

4—2 



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52 The Indejinahles of Mathematics [chap. lY 

the relation affirmed between A and B in the proposition " A differs 
from 5" is the general relation of difference, and is precisely and 
numerically the same as the relation affirmed between C and Z) in 
"C differs from Z>." And this doctrine must be held, for the same 
reasons, to be true of all other relations ; relations do not have instances, 
but are strictly the same in all propositions in which they occur. 

We may now sum up the main points elicited in our discussion of 
the verb. The verb, we saw, is a concept which, like the adjective, may 
occur in a proposition without being one of the terms of the proposition, 
though it may also be made into a logical subject. One verb, and one 
only, must occur as verb in every proposition ; but every proposition, 
by turning its verb into a verbal noun, can be changed into a single 
logical subject, of a kind which I shall call in future a propositional 
concept. Every verb, in the logical sense of the word, may be regarded 
as a relation ; when it occurs as verb, it actually relates, but when it 
occurs as verbal noun it is the bare relation considered independently of 
the terms which it relates. Verbs do not, like adjectives, have instances, 
but are identical in all the cases of their occurrence. Owing to the way 
in which the verb actually relates the terms of a proposition, every 
proposition has a unity which renders it distinct from the sum of its 
constituents. All these points lead to logical problems, which, in a 
treatise on logic, would deserve to be fully and thoroughly discussed. 

Having now given a general sketch of the nature of verbs and 
adjectives, I shall proceed, in the next two chapters, to discussions 
arising out of the consideration of adjectives, and in Chapter vii to 
topics connected with verbs. Broadly speaking, classes are connected 
with adjectives, while propositional functions involve verbs. It is for 
this reason that it has been necessary to deal at such length with a 
subject which might seem, at first sight, to be somewhat remote from 
the principles of mathematics. 

Aristotelian Society, 1900 — 1901) must not be applied to all concepts. The relation of 
an instance to its universal-, at any rate^ must be actually and numerically the same 
in all cases where it occurs. 



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CHAPTER V. 



DENOTING. 



56. The notion of denoting, like most of the notions of logic, has 
been obscured hitherto by an undue admixture of psychology. There is 
a sense in which we denote, when we point or describe, or employ words 
as symbols for concepts ; this, however, is not the sense that I wish to 
discuss. But the fact that description is possible — that we are able, by 
the employment of concepts, to designate a thing which is not a concept 
— is due to a logical relation between some concepts and some terms, in 
virtue of which such concepts inherently and logically denote such terms. 
It is this sense of denoting which is here in question. This notion lies 
at the bottom (I think) of all theories of substance, of the subject- 
predicate logic, and of the opposition between things and ideas, 
discursive thought and immediate perception. These various develop- 
ments, in the main, appear to me mistaken, while the fundamental fact 
itself, out of which they have grown, is hardly ever discussed in its 
logical purity. 

A concept denotes when, if it occurs in a proposition, the proposition 
is not about the concept, but about a term connected in a certain 
peculiar way with the concept. If I say " I met a man," the proposition 
is not about a man: this is a concept which does not walk the streets, 
but lives in the shadowy limbo of the logic-books. What I met was a 
thing, not a concept, an actual man with a tailor and a bank-account or 
a public-house and a drunken wife. Again, the proposition "any finite 
number is odd or even '' is plainly true ; yet the concept " any finite 
number " is neither odd nor even. It is only particular numbers that are 
odd or even ; there is not, in addition to these, another entity, any 
number, which is either odd or even, and if there were, it is plain that it 
could not be odd and could not be even. Of the concept "any number,'' 
almost all the propositions that contain the phrase " any number '' are 
false. If we wish to speak of the concept, we have to indicate the fact by 
italics or inverted commas. People often assert that man is mortal ; 
but what is mortal will die, and yet we should be surprised to find in the 
"Times'' such a notice as the following: "Died at his residence of 



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54 The Indefinahles of Mathematics [chap, v 

Camelot, Gladstone Road, Upper Tooting, on the 18th of June 19 — , 
Man, eldest son of Death and Sin."' Man^ in fact, does not die ; hence 
if " man is mortal *" were, as it appears to be, a proposition about man^ 
it would be simply false. The fact is, the proposition is about men ; 
and here again, it is not about the concept men^ but about what this 
concept denotes. The whole theory of definition, of identity, of classes, 
of symbolism, and of the variable is wrapped up in the theory of 
denoting. The notion is a fundamental notion of logic, and, in spite 
of its difficulties, it is quite essential to be as clear about it as possible. 

57. The notion of denoting may be obtained by a kind of logical 
genesis from subject-predicate propositions, upon which it seems more or 
less dependent. The simplest of propositions are those in which one 
predicate occurs otherwise than as a term, and there is only one term of 
which the predicate in question is asserted. Such propositions may be 
called subject-predicate propositions. Instances are: A is, A is one, 
A is human. Concepts which are predicates might also be called class- 
concepts, because they give rise to classes, but we shall find it necessary 
to distinguish between the -wovA.^ predicate and class-concept. Propositions 
of the subject -predicate type always imply and are implied by other propo- 
sitions of the type which asserts that an individual belongs to a class. 
Thus the above instances are equivalent to : A is an entity, ^ is a unit, 
^ is a man. These new propositions are not identical with the previous 
ones, since they have an entirely different form. To begin with, is is now 
the only concept not used as a term. A man^ we shall find, is neither 
a concept nor a term, but a certain kind of combination of certain terms, 
namely of those which are human. And the relation of Socrates to 
a man is quite different from his relation to humanity ; indeed "Socrates 
is human ^' must be held, if the above view is correct, to be not, in the 
most usual sense, a judgment of relation between Socrates and humanity, 
since this view would make human occur as term in "Socrates is human." 
It is, of course, undeniable that a relation to humanity is implied by 
•' Socrates is human,'' namely the relation expressed by " Socrates has 
humanity''; and this relation conversely implies the subject-predicate 
proposition. But the two propositions can be clearly distinguished, and 
it is important to the theory of classes that this should be done. Thus 
we have, in the case of every predicate, three types of propositions 
which imply one another, namely, "Socrates is human," "Socrates has 
humanity," and ^'Socrates is a man." The first contains a term and 
a predicate, the second two terms and a relation (the second term being 
identical with the predicate of the first proposition)*, while the third 
contains a term, a relation, and what I shall call a disjunction (a term 
which will be explained shortly)-)*. The class-concept differs little, if at 

•^ Cf. § 49. 

t There are two allied propositions expressed by the same words^ namely 
^^ Socrates is a-man" and ^^^ Socrates is-a man." The above remarks apply to the 



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56-58] Denoting 55 

all, from the predicate, while the class, as opposed to the class-concept, is 
the sum or conjunction of all the terms which have the given predicate. 
The relation which occurs in the second type (Socrates has humanity) is 
characterized completely by the fact that it implies and is implied by a 
proposition with only one term, in which the other term of the relation 
has become a predicate. A class is a certain combination of terms, a 
class-concept is closely akin to a predicate, and the terms whose com- 
bination forms the class are determined by the class-concept. Predicates 
are, in a certain sense, the simplest type of concepts, since they occur in 
the simplest type of proposition. 

58. There is, connected with every predicate, a great variety of 
closely allied concepts, which, in so far as they are distinct, it is 
important to distinguish. Starting, for example, with Jiuman^ we have 
man, men, all men, every man, any man, the human race, of which all 
except the first are twofold, a denoting concept and an object denoted ; 
we have also, less closely analogous, the notions " a man ^' and " some 
man,^' which again denote objects* other than themselves. This vast 
apparatus connected with every predicate must be borne in mind, and 
an endeavour must be made to give an analysis of all the above notions. 
But for the present, it is the property of denoting, rather than the 
various denoting concepts, that we are concerned with. 

The combination of concepts as such to form new concepts, of greater 
complexity than their constituents, is a subject upon which writers on 
logic have said many things. But the combination of terms as such, 
to form what by analogy may be called complex terms, is a subject 
upon which logicians, old and new, give us only the scantiest discussion. 
Nevertheless, the subject is of vital importance to the philosophy of 
mathematics, since the nature both of number and of the variable turns 
upon just this point. Six words, of constant occurrence in daily life, 
are also characteristic of mathematics : these are the words all^ every ^ 
any^ a, some and the. For correctness of reasoning, it is essential that 
these words should be sharply distinguished one from another; but 
the subject bristles with difficulties, and is almost wholly neglected by 
logicians f. 

It is plain, to begin with, that a phrase containing one of the above 

former ; but in future^ unless the contrary is indicated by a hyphen or otherwise_, 
the latter will always be in question. The former expresses the identity of Socrates 
with an ambiguous individual ; the latter expresses a relation of Socrates to the 
class-concept man. 

* I shall use the word object in a wider sense than term^ to cover both singular 
and plural^ and also cases of ambiguity^ such as ^'^a man." The fact that a word can 
be framed with a wider meani. ^ than term raises grave logical problems. Cf. § 47. 

t On the indefinite articiv^.^, some good remarks are made by Meinong, 
^''Abstrahiren und Vergleichen/' Zeitschrift fur Psychologie und Physiologie der 
Sinnesorgane, Vol. xxiv, p. 63. 



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56 The Indefinahles of Mathematics [chap, y 

six words always denotes. It will be convenient, for the present 
discussion, to distinguish a class-concept from a predicate : I shall call 
human a predicate, and man a class-concept, though the distinction is 
perhaps only verbal. The characteristic of a class-concept, as distin- 
guished from terras in general, is that " x is a u '' is a propositional 
function when, and only when, u is a class-concept. It must be held that 
when u \^ not a class-concept, we do not have a false proposition, but 
simply no proposition at all, whatever value we may give to x. This 
enables us to distinguish a class-concept belonging to the null-class, for 
which all propositions of the above form are false, from a term which is 
not a class-concept at all, for which there are no propositions of the 
above form. Also it makes it plain that a class-concept is not a term 
in the proposition '' x is a 2^,^' for u has a restricted variability if the 
formula is to remain a proposition. A denoting phrase, we may now say, 
consists always of a class-concept preceded by one of the above six words 
or some synonym of one of them. 

59. The question which first meets us in regard to denoting is 
this : Is there one way of denoting six different kinds of objects, or are 
the ways of denoting different ? And in the latter case, is the object 
denoted the same in all six cases, or does the object differ as well as the 
way of denoting it ? In order to answer this question, it will be first 
necessary to explain the differences between the six words in question. 
Here it will be convenient to omit the word the to begin with, since this 
word is in a different position from the others, and is liable to limitations 
from which they are exempt. 

In cases where the class defined by a class-concept has only a finite 
number of terms, it is possible to omit the class-concept wholly, and 
indicate the various objects denoted by enumerating the terms and 
connecting them by means of and or or as the case may be. It will 
help to isolate a part of our problem if we first consider this case, 
although the lack of subtlety in language renders it difficult to grasp the 
difference between objects indicated by the same form of words. 

Let us begin by considering two terms only, say Brown and Jones. 
The objects denoted by all^ every ^ any^ a and some^ are respectively 
involved in the following five propositions. (1) Brown and Jones are 
two of Miss Smith's suitors ; (^) Brown and Jones are paying court to 
Miss Smitin (3) if it was Brown or Jones you met, it was a very ardent 
lover; (4) if it was one of Miss Smith's suitors, it must have been 
Brown or Jones ; (5) Miss Smith will marry Brown or Jones. Although 
only two forms of words. Brown and Jones and Broion or Jones^ are 
involved in these propositions, I maintain that five different combinations 
are involved. The distinctions, some of which eve rather subtle, may be 

* I intend to distinguisli between a and some in ? way not warranted by language; 
the distinction of all and every is also a straining of usage. Both are necessary to 
avoid circumlocution. 



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58, 59] Denoting 57 

brought out by the following considerations. In the first proposition, it 
is Brown and Jones who are two, and this is not true of either separately; 
nevertheless it is not the whole composed of Brown and Jones which is 
two, for this is only one. The two are a genuine combination of Brown 
with Jones, the kind of combination which, as we shall see in the next 
chapter, is characteristic of classes. In the second proposition, on the 
contrary, what is asserted is true of Brown and Jones severally; the 
proposition is equivalent to, though not (I think) identical with, "Brown 
is paying court to Miss Smith afid Jones is paying court to Miss Smith." 
Thus the combination indicated by and is not the same here as in the 
first case: the first case concerned all of them collectively, while the 
second concerns all distributively, i.e, each or every one of them. For 
the sake of distinction, we may call the first a niimerical conjunction, 
since it gives rise to number, the second a propositional conjunction, 
since the proposition in which it occurs is equivalent to a conjunction of 
propositions. (It should be observed that the conjunction of propo- 
sitions in question is of a wholly different kind from any of the com- 
binations we are considering, being in fact of the kind which is called 
the logical product. The propositions are combined qua propositions, 
not qua terms.) 

The third proposition gives the kind of conjunction by which any is 
defined. There is some difficulty about this notion, which seems half-way 
between a conjunction and a disjunction. This notion may be further 
explained as follows. Let a and h be two diff'erent propositions, 
each of which implies a third proposition c. Then the disjunction 
"a or 6" implies c. Now let a and b be propositions assigning the 
same predicate to two different subjects, then there is a combination 
of the two subjects to which the given predicate may be assigned so 
that the resulting proposition is equivalent to the disjunction "a or 6." 
Thus suppose we have " if you met Brown, you met a very ardent lover,**' 
and " if you met Jones, you met a very ardent lover.'*'' Hence we infer 
" if you met Brown or if you met Jones, you met a very ardent lover,'" 
and we regard this as equivalent to " if you met Brown or Jones, etc.'''' 
The combination of Brown and Jones here indicated is the same as that 
indicated hy either of them. It differs from a disjunction by the fact 
that it implies and is implied by a statement concerning both ; but in 
some more complicated instances, this mutual implication fails. The 
method of combination is, in fact, different from that indicated by both^ 
and is also different from both forms of disjunction. I shall call it the 
variable conjunction. The first form of disjunction is given by (4) : this 
is the form which I shall denote by a suitor. Here, although it must 
have been Brown or Jones, it is not true that it must have been Brown, 
nor yet that it must have been Jones. Thus the proposition is not 
equivalent to the disjunction of propositions "it must have been Brown 
or it must have been Jones,'''' The proposition, in fact, is not capable of 



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58 The Indefinables of Mathematics [chap, y 

statement either as a disjunction or as a conjunction of propositions, 
except in the very roundabout form : " if it was not Brown, it was 
Jones, and if it was not Jones, it was Brown,^' a form which rapidly 
becomes intolerable when the number of terms is increased beyond two, 
and becomes theoretically inadmissible when the number of terms is 
infinite. Thus this form of disjunction denotes a variable term, that 
is, whichever of the two terms we fix upon, it does not denote this term, 
and yet it does denote one or other of them. This form accordingly I 
shall call the variable disjunction. Finally, the second form of disjunction 
is given by (5). This is what I shall call the constant disjunction, since 
here either Brown is denoted, or Jones is denoted, but the alternative 
is undecided. That is to say, our proposition is now equivalent to a 
disjunction of propositions, namely " Miss Smith will marry Brown, or 
she will marry Jones. '^ She will marry some one of the two, and the 
disjunction denotes a particular one of them, though it may denote 
either particular one. Thus all the five combinations are distinct. 

It is to be observed that these five combinations yield neither terms 
nor concepts, but strictly and only combinations of terms. The first 
yields many terms, while the others yield something absolutely peculiar, 
which is neither one nor many. The combinations are combinations of 
terms, effected without the use of relations. Corresponding to each 
combination there is, at least if the terms combined form a class, a 
perfectly definite concept, which denotes the various terms of the combi- 
nation combined in the specified manner. To explain this, let us repeat 
our distinctions in a case where the terms to be combined are not 
enumerated, as above, but are defined as the terms of a certain class. 

60. When a class-concept a is given, it must be held that the 
various terms belonging to the class are also given. That is to say, any 
term being proposed, it can be decided whether or not it belongs to the 
class. In this way, a collection of terms can be given otherwise than by 
enumeration. Whether a collection can be given otherwise than by 
enumeration or by a class -concept, is a question which, for the present, 
I leave undetermined. But the possibility of giving a collection by a 
class-concept is highly important, since it enables us to deal with infinite 
collections, as we shall see in Part V. For the present, I wish to examine 
the meaning of such phrases as all a\ every a, any «, an <x, and some a. 
All a\ to begin with, denotes a numerical conjunction; it is definite as 
soon as a is given. The concept all «'s is a perfectly definite single 
concept, which denotes the terms of a taken all together. The terms 
so taken have a number, which may thus be regarded, if we choose, as 
a property of the class-concept, since it is determinate for any given 
class -concept. Every a, on the contrary, though it still denotes all the 
a's, denotes them in a different way, ix. severally instead of collectively. 
Any a denotes only one a, but it is wholly irrelevant which it denotes, 
and what is said will be equally true whichever it may be. Moreover, 



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59-61] Denoting 59 

any a denotes a variable a^ that is, whatever particular a we may fasten 
upon, it is certain that any a does not denote that one ; and yet of that 
one any proposition is true which is true of any a. An a denotes a 
variable disjunction : that is to say, a proposition which holds of an a 
may be false concerning each particular «, so that it is not reducible to 
a disjunction of propositions. For example, a point lies between any 
point and any other point ; but it would not be true of any one 
particular point that it lay between any point and any other point, 
since there would be many pairs of points between which it did not lie. 
This brings us finally to some a, the constant disjunction. This denotes 
just one term of the class a^ but the term it denotes may be any term 
of the class. Thus " some moment does not follow any moment " would 
mean that there was a first moment in time, while " a moment precedes 
any moment ^' means the exact opposite, namely, that every moment has 
predecessors, 

61. In the case of a class a which has a finite number of terms — 
say a^^ a.^^ a.^^ , ... a.^^ we can illustrate these various notions as follows : 

(1) All a's denotes a^ and a.^ and ... and a^. 

(^) Every a denotes a^ and denotes a<^ and . . . and denotes a^. 

(3) Any a denotes a^ or a^ or ... or <2^, where or has the meaning 
that it is irrelevant which we take. 

(4) An a denotes a^ or a. or ... or an^ where or has the meaning 
that no one in particular must be taken, just as in all a''^ we must not 
take any one in particular. 

(5) Some a denotes a^ or denotes a^ or ... or denotes a^, where it is 
not irrelevant which is taken, but on the contrary some one particular a 
must be taken. 

As the nature and properties of the various ways of combining terms 
are of vital importance to the principles of mathematics, it may be well 
to illustrate their properties by the following important examples. 

{a) Let ^ be a class, and h a class of classes. We then obtain 
in all six possible relations oi a to h from various combinations of any^ 
a and some. All and every do not, in this case, introduce anything new. 
The six cases are as follows. 

(1) Any a belongs to any class belonging to 6, in other words, the 
class a is wholly contained in the common part or logical product of 
the various classes belonging to h. 

(S) Any a belongs to a h^ i.e. the class a is contained in any 
class which contains all the Z^'s, or, is contained in the logical sum of 
all the 6's. 

(3) Any a belongs to some &, i.e. there is a class belonging to 6, 
in which the class a is contained. The difference between this case and 
the second arises from the fact that here there is one h to which every 
a belongs, whereas before it was only decided that every a belonged to 
a ^, and different ^'s might belong to different h\. 



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60 The Indefinables of Mathematics [ghap, v 

(4) An a belongs to any b^ i.e. whatever h we take, it has a part 
in common with a. 

(5) An a belongs to a b^ i.e. there is a Z? which has a part in common 
with a. This is equivalent to " some (or an) a belongs to some Z?.'' 

(6) Some a belongs to any 5, i.e. there is an a which belongs to 
the common part of all the 5's, or a and all the b\ have a common part. 
These are all the cases that arise here. 

(/3) It is instructive, as showing the generality of the type of 
relations here considered, to compare the above case with the following. 
Let <2, b be two series of real numbers ; then six precisely analogous 
cases arise. 

(1) Any a is less than any Z?, or, the series a is contained among 
numbers less than every b. 

(2) Any a is less than a b^ or, whatever a we take, there is a Z? 
which is greater, or, the series a is contained among numbers less than 
a (variable) term of the series b. It does not follow that some term of 
the series b is greater than all the a"^. 

(3) Any a is less than some b^ or, there is a term of b which is 
greater than all the a\. This case is not to be confounded with (2). 

(4) An a is less than any 5, i.e. whatever b we take, there is an 
a which is less than it. 

(5) An a is less than a b^ i.e. it is possible to find an a and a b 
such that the a is less than the b. This merely denies that any a is 
greater than any b. 

(6) Some a is less than any 6, i.e. there is an a which is less than 
all the F^. This was not implied in (4), where the a was variable, 
whereas here it is constant. 

In this case, actual mathematics have compelled the distinction 
between the variable and the constant disjunction. But in other cases, 
where mathematics have not obtained sway, the distinction has been 
neglected ; and the mathematicians, as was natural, have not investi- 
gated the logical nature of the disjunctive notions which they employed. 

(7) I shall give one other instance, as it brings in the difference 
between any and every^ which has not been relevant in the previous 
cases. Let a and b be two classes of classes ; then twenty different 
relations between them arise from different combinations of the terms 
of their terms. The following technical terms will be useful. If a be 
a class of classes, its logical sum consists of all terms belonging to any 
^, i.e. all terms such that there is an a to which they belong, while 
its logical product consists of all terms belonging to every a, i.e. to the 
common part of all the ^^'s. We have then the following cases. 

(1) Any term of any a belongs to every 6, i.e. the logical sum of 
a is contained in the logical product of b. 

(2) Any term of any a belongs to a &, i.e. the logical sum of a 
is contained in the logical sum of b. 



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61, 62] Denoting 61 

(3) Any term of any a belongs to some /;, i.e. there is a h whicli 
contains the logical sum of a. 

(4) Any term of some (or an) a belongs to every 5, i.e. there is an 
a which is contained in the product of 5. 

(5) Any term of some (or an) a belongs to a 5, i.e. there is an a 
which is contained in the sum oih. 

(6) Any term of some (or an) a belongs to some 5, i.e. there is a 
h which contains one class belonging to a. 

(7) A term of any a belongs to any h^ i.e. any class of a and any 
class of h have a common part. 

(8) A term of any a belongs to a h^ i.e. any class of a has a part 
in common with the logical sum of h. 

(9) A term of any a belongs to some 5, i.e. there is a h with which 
any a has a part in common. 

(10) A term of an a belongs to every 5, i.e. the logical sum of a 
and the logical product of h have a common part. 

(11) A term of an a belongs to any h^ i.e. given any h^ an a can 
be found with which it has a common part. 

(IS) A term of an a belongs to a 5, i.e. the logical sums of a and 
of h have a common part. 

(13) Any term of every a belongs to every b, i.e. the logical 
product of a is contained in the logical product of b. 

(14) Any term of every a belongs to a 5, i.e. the logical product 
of 61 is contained in the logical sum of b. 

(15) Any term of every a belongs to some b^ i.e. there is a term 
of b in which the logical product of a is contained. 

(16) A (or some) term of every a belongs to every b, i.e. the logical 
products of a and of b have a common part. 

(17) A (or some) term of every a belongs to a b^ i.e. the logical 
product of a and the logical sum of b have a common part. 

(18) Some term of any a belongs to every 5, i.e. any a has a part 
in common with the logical product of b. 

(19) A term of some a belongs to any 6, i.e. there is some term 
of a with which any b has a common part. 

(SO) A term of every a belongs to any b, i.e. any b has a part in 
common with the logical product of a. 

The above examples show that, although it may often happen that 
there is a mutual implication (which has not always been stated) of 
corresponding propositions concerning some and a, or concerning any 
and every, yet in other cases there is no such mutual implication. Thus 
the five notions discussed in the present chapter are genuinely distinct^ 
and to confound them may lead to perfectly definite fallacies. 

62. It appears from the above discussion that, whether there are 
different ways of denoting or not, the objects denoted by all men, every 
man, etc. are certainly distinct. It seems therefore legitimate to say 



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62 The Indejinables of Mathematics [chap, v 

that the whole difference Hes in the objects, and that denoting itself is 
the same in all cases. There are, however, many difficult problems 
connected with the subject, especially as regards the nature of the 
objects denoted. All men^ which I shall identify with the class of men, 
seems to be an unambiguous object, although grammatically it is plural. 
But in the other cases the question is not so simple : we may doubt 
whether an ambiguous object is unambiguously denoted, or a definite 
object ambiguously denoted. Consider again the proposition " I met 
a man." It is quite certain, and is implied by this proposition, that 
what I met was an unambiguous perfectly definite man : in the technical 
language which is here adopted, the proposition is expressed by " I met 
some man.'" But the actual man whom I met forms no part of the 
proposition in question, and is not specially denoted by some man. 
Thus the concrete event which happened is not asserted in the proposi- 
tion. What is asserted is merely that some one of a class of concrete 
events took place. The whole human race is involved in my assertion : 
if any man who ever existed or will exist had not existed or been going 
to exist, thfe purport of my proposition would have been different. Or, 
to put the same point in more intensional language, if I substitute for 
man any of the other class-concepts applicable to the individual whom 
I had the honour to meet, my proposition is changed, although the 
individual in question is just as much denoted as before. What this 
proves is, that some man must not be regarded as actually denoting 
Smith and actually denoting Brown, and so on : the whole procession 
of human beings throughout the ages is always relevant to every pro- 
position in which sovie man occurs, and what is denoted is essentially 
not each separate man, but a kind of combination of all men. This 
is more evident in the case of every ^ any^ and a. There is, then, a 
definite something, different in each of the five cases, which must, in 
a sense, be an object, but is characterized as a set of terms combined 
in a certain way, which something is denoted by all men^ every man^ 
any man^ a man or some man ; and it is with this very paradoxical 
object that propositions are concerned in which the corresponding 
concept is used as denoting. 

63. It remains to discuss the notion of the. This notion has 
been symbolically emphasized by Peano, with very great advantage to 
his calculus; but here it is to be discussed philosophically. The use 
of identity and the theory of definition are dependent upon this notion, 
which has thus the very highest philosophical importance. 

The word the^ in the singular, is correctly employed only in relation 
to a class-concept of which there is only one instance. We speak of 
the King, the Prime Minister, and so on (understanding at the present 
time) ; and in such cases there is a method of denoting one single definite 
term by means of a concept, which is not given us by any of our other five^ 
words. It is owing to this notion that mathematics can give definitions 



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62-64] Denoting 63 

of terms which are not concepts— a possibiUty which illustrates the 
difference between mathematical and philosophical definition. Every 
term is the only instance of some class-concept, and thus every term, 
theoretically, is capable of definition, provided we have not adopted 
a system in which the said term is one of our indefinables. It is a 
curious paradox, puzzling to the symbolic mind, that definitions, theo- 
retically, are nothing but statements of symbolic abbreviations, irrelevant 
to the reasoning and inserted only for practical convenience, while yet, 
in the development of a subject, they always require a very large amount 
of thought, and often embody some of the greatest achievements of 
analysis. This fact seems to be explained by the theory of denoting. 
An object may be present to the mind, without our knowing any concept 
of which the said object is the instance; and the discovery of such a 
concept is not a mere improvement in notation. The reason why this 
appears to be the case is that, as soon as the definition is found, it 
becomes wholly unnecessary to the reasoning to remember the actual 
object defined, since only concepts are relevant to our deductions. In 
the moment of discovery, the definition is seen to be ^rz^£^, because the 
object to be defined was already in our thoughts ; but as part of our 
reasoning it is not true, but merely symbolic, since what the reasoning 
requires is not that it should deal with that object, but merely that 
it should deal with the object denoted by the definition. 

In most actual definitions of mathematics, what is defined is a class 
of entities, and the notion of the does not then explicitly appear. But 
even in this case, what is really defined is the class satisfying certain 
conditions ; for a class, as we shall see in the next chapter, is always 
a term or conjunction of terms and never a concept. Thus the notion of 
the is always relevant in definitions ; and we may observe generally that 
the adequacy of concepts to deal with things is wholly dependent upon 
the unambiguous denoting of a single term which this notion gives. 

64. The connection of denoting with the nature of identity is 
important, and helps, I think, to solve some rather serious problems. 
The question whether identity is or is not a relation, and even whether 
there is such a concept at all, is not easy to answer. For, it may be 
said, identity cannot be a relation, since, where it is truly asserted, 
we have only one term, whereas two terms are required for a relation. 
And indeed identity, an objector may urge, cannot be anything at all : 
two terms plainly are not identical, and one term cannot be, for what 
is it identical with.? Nevertheless identity must be something. We 
might attempt to remove identity from terms to relations, and say that 
two terms are identical in some respect when they have a given relation 
to a given term. But then we shall have to hold either that there is 
strict identity between the two cases of the given relation, or that the 
two cases have identity in the sense of having a given relation to a given 
term ; but the latter view leads to an endless process of the illegitimate 



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64 The Indefinahles of Mathematics [chap, v 

kind. Thus identity must be admitted, and the difficulty as to the 
two terms of a relation must be met by a sheer denial that two different 
terms are necessary. There must always be a referent and a relatum, 
but these need not be distinct ; and where identity is affirmed, they are 
not so*. 

But the question arises : Why is it ever worth while to affirm 
identity.^ This question is answered by the theory of denoting. If 
we say " Edward VII is the King,"' we assert an identity ; the reason 
why this assertion is worth making is, that in the one case the actual 
term occurs, while in the other a denoting concept takes its place. 
(For purposes of discussion, I ignore the fact that Edwards form a 
class, and that seventh Edwards form a class having only one term. 
Edward VII is practically, though not formally, a proper name.) Often 
two denoting concepts occur, and the term itself is not mentioned, as 
in the proposition "the present Pope is the last survivor of his genera- 
tion.'' V^^hen a term is given, the assertion of its identity with itself, 
though true, is perfectly futile, and is never made outside the logic- 
books ; but where denoting concepts are introduced, identity is at once 
seen to be significant. In this case, of course, there is involved, though 
not asserted, a relation of the denoting concept to the term, or of the 
two denoting concepts to each other. But the is which occurs in such 
propositions does not itself state this further relation, but states pure 
identity]*. 

65. To sum up. When a class-concept, preceded by one of the 
six words all^ every^ any^ «, some^ the^ occurs in a proposition, the 
proposition is, as a rule, not about the concept formed of the two words, 
together, but about an object quite different from this, in general not 
a concept at all, but a term or complex of terms. This may be seen by 
the fact that propositions in which such concepts occur are in general 
false concerning the concepts themselves. At the same time, it is 
possible to consider and make propositions about the concepts them- 
selves, but these are not the natural propositions to make in employing 
the concepts. " Any number is odd or even " is a perfectly natural propo- 
sition, whereas ''Any number is a variable conjunction'' is a proposition 
only to be made in a logical discussion. In such cases, we say that the 
concept in question denotes. We decided that denoting is a perfectly 

■^ On relations of terms to tliemselves_, v, inf. Chap, ix^ § 95. 

t The word u is terribly ambiguous^ and great care is necessary in order not to 
confound its various meanings. We have (1) the sense in which it asserts Beings as 
in '^ A is"; (2) the sense of identity; (3) the sense of predication^ in '' A is human"; 
(4) the sense of '^ A is a-man" (cf. p. 54^ note), which is very like identity. In 
addition to these there are less common uses^, as ^'^to be good is to be happy ^" where 
a relation of assertions is meant, that relation^ in fact, which, where it exists^ gives 
rise to formal implication. Doubtless there are further meanings which have not 
occurred to me. On the meanings of is, cf. De Morgan, Formal Logic, pp. 49, 50. 



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64, 65] Denoting 65 

definite relation, the same in all six cases, and that it is the nature of 
the denoted object and the denoting concept which distinguishes the 
cases. We discussed at some length the nature and the differences of 
the denoted objects in the five cases in which these objects are com- 
binations of terms. In a full discussion, it would be necessary also to 
discuss the denoting concepts : the actual meanings of these concepts, as 
opposed to the nature of the objects they denote, have not been discussed 
above. But I do not know that there would be anything further to say 
on this topic. Finally, we discussed the^ and showed that this notion 
is essential to what mathematics calls definition, as well as to the 
possibility of uniquely determining a term by means of concepts ; the 
actual use of identity, though not its meaning, was also found to depend 
upon this way of denoting a single term. From this point we can 
advance to the discussion of classes, thereby continuing the development 
of the topics connected with adjectives. 



E. 



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CHAPTER VI. 

CLASSES. 

66. To bring clearly before the mind what is meant by class ^ and 
to distinguish this notion from all the notions to which it is allied, is 
one of the most difficult and important problems of mathematical 
philosophy. Apart from the fact that class is a very fundamental 
concept, the utmost care and nicety is required in this subject on 
account of the contradiction to be discussed in Chapter x. I must 
ask the reader, therefore, not to regard as idle pedantry the apparatus 
of somewhat subtle discriminations to be found in what follows. 

It has been customary, in works on logic, to distinguish two stand- 
points, that of extension and that of intension. Philosophers have 
usually regarded the latter as more fundamental, while Mathematics 
has been held to deal specially with the former. M. Couturat, in his 
admirable work on Leibniz, states roundly that Symbolic Logic can only 
be built up from the standpoint of extension* ; and if there really were 
only these two points of view, his statement would be justified. But as 
a matter of fact, there are positions intermediate between pure intension 
and pure extension, and it is in these intermediate regions that Symbolic 
Logic has its lair. It is essential that the classes with which we are 
concerned should be composed of terms, and should not be predicates or 
concepts, for a class must be definite when its terms are given, but 
in general there will be many predicates which attach to the given 
terms and to no others. We cannot of course attempt an intensional 
definition of a class as the class of predicates attaching to the terms 
in question and to no others, for this would involve a vicious circle ; 
hence the point of view of extension is to some extent unavoidable. 
On the other hand, if we take extension pure, our class is defined by 
enumeration of its terms, and this method will not allow us to deal, as 
Symbolic Logic does, with infinite classes. Thus our classes must in 
general be regarded as objects denoted by concepts, and to this extent 
the point of view of intension is essential. It is owing to this con- 

* La Logique de Leibni^y Paris, 1901^ p. 387. 



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66-68] Classes 67 

sideration that the theory of denoting is of such great importance. In 
the present chapter we have to specify the precise degree in which 
extension and intension respectively enter into the definition and etn- 
ployment of classes ; and throughout the discussion, I must ask the 
reader to remember that whatever is said has to be applicable to infinite 
as well as to finite classes. 

67. When an object is unambiguously denoted by a concept, I shall 
speak of the concept as a concept (or sometimes, loosely, as the concept) 
of the object in question. Thus it will be necessary to distinguish the 
concept of a class from a class-concept. We agreed to call mari a class- 
concept, but man does not, in its usual employment, denote anything. 
On the other hand, men and all men (which I shall regard as synonyms) do 
denote, and I shall contend that what they denote is the class composed 
of all men. Thus man is the class-concept, vien (the concept) is the 
concept of the class, and men (the object denoted by the concept vien) 
are the class. It is no doubt confusing, at first, to use elass-concept and 
concept of a class in different senses ; but so many distinctions are 
required that some straining of language seems unavoidable. In 
the phraseology of the preceding chapter, we may say that a class is a 
numerical conjunction of terms. This is the thesis which is to be 
established. 

68. In Chapter ii we regarded classes as derived from assertions, 
ix, as all the entities satisfying some assertion, whose form was left 
wholly vague. I shall discuss this view critically in the next chapter ; 
for the present, we may confine ourselves to classes as they are derived 
from predicates, leaving open the question whether every assertion is 
equivalent to a predication. We may, then, imagine a kind of genesis 
of classes, through the successive stages indicated by the typical propo- 
sitions " Socrates is human,"' " Socrates has humanity," " Socrates is a 
man,'' " Socrates is one among men." Of these propositions, the last 
only, we should say, explicitly contains the class as a constituent ; but 
every subject-predicate proposition gives rise to the other three eiquivalent 
propositions, and thus every predicate (provided it can be sometimes 
truly predicated) gives rise to a class. This is the genesis of classes from 
the intensional standpoint. 

On the other hand, when mathematicians deal with what they call a 
manifold, aggregate, Menge, ensemble, or some equivalent name, it is 
common, especially where the number of terms involved is finite, to regard 
the object in question (which is in fact a class) as defined by the enumera- 
tion of its terms, and as consisting possibly of a single term, which in 
that case is the class. Here it is not predicates and denoting that are 
relevant, but terms connected by the word and, in the sense in which 
this word stands for a numerical conjunction. Thus Brown and Jones 
are a class, and Brown singly is a class. This is the extensional genesis 
of classes. 

5—2 



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68 The Indejinables of Mathematics [chap, vi 

69. The best formal treatment of classes in existence is that of 
Peano*. But in this treatment a number of distinctions of great 
philosophical importance are overlooked. Peano, not I think quite 
consciously, identifies the class with the class-concept ; thus the relation 
of an individual to its class is, for him, expressed by is a. For him, 
" 2 is a number '"* is a proposition in which a term is said to belong to 
the class number. Nevertheless, he identifies the equality of classes, 
which consists in their having the same terms, with identity — a pro- 
ceeding which is quite illegitimate when the class is regarded as the 
class-concept. In order to perceive that man and featJierless biped are 
not identical, it is quite unnecessary to take a hen and deprive the poor 
bird of its feathers. Or, to take a less complex instance, it is plain that 
even prime is not identical with integer next after 1. Thus when we 
identify the class with the class-concept, we must admit that two classes 
may be equal without being identical. Nevertheless, it is plain that 
when two class-concepts are equal, some identity is involved, for we say 
that they have the same terms. Thus there is some object which is 
positively identical when two class-concepts are equal ; and this object, 
it would seem, is more properly called the class. Neglecting the plucked 
hen, the class of featherless bipeds, every one would say, is the same as 
the class of men ; the class of even primes is the same as the class of 
integers next after 1. Thus we must not identify the class with the 
class-concept, or regard " Socrates is a man " as expressing the relation 
of an individual to a class of which it is a member. This has two 
consequences (to be established presently) which prevent the philosophi(i|^l 
acceptance of certain points in Peano's formalism. The first consequence 
is, that there is no such thing as the null-class, though there are null 
class-concepts. The second is, that a class having only one term is to 
be identified, contrary to Peano's usage, with that one term. I should 
not propose, however, to alter his practice or his notation in consequence 
of either of these points ; rather I should regard them as proofs that 
Symbolic Logic ought to concern itself, as /ar as notation goes, with 
class-concepts rather than with classes. 

70. A class, we have seen, is neither a predicate nor a class- 
concept, for diflPerent predicates and different class-concepts may corre- 
spond to the same class. A class also, in one sense at least, is distinct 
from the whole composed of its terms, for the latter is only and essentially 
one, while the former, where it has many terms, is, as we shall see later, 
the very kind of object of which many is to be asserted. The distinction 
of a class as many from a class as a whole is often made by language : 
space and points, time and instants, the army and the soldiers, the navy 
and the sailors, the Cabinet and the Cabinet Ministers, all illustrate the 
distinction. The notion of a whole, in the sense of a pure aggregate 

* Neglecting Frege^ who is discussed in the Appendix. 



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69-71] Classes 69 

which is here relevant, is, we shall find, not always applicable where the 
notion of the class as many applies (see Chapter x). In such cases, 
though terms may be said to belong to the class, the class must not be 
treated as itself a single logical subject*. But this case never arises 
where a class can be generated by a predicate. Thus we may for the 
present dismiss this complication from our minds. In a class as many, 
the component terms, though they have some kind of unity, have less 
than is required for a whole. They have, in fact, just so much unity 
as is required to make them many, and not enough to prevent them from 
remaining many. A further reason for distinguishing wholes from 
classes as many is that a class as one may be one of the terms of itself 
as many, as in " classes are one among classes " (the extensional equi- 
valent of " class is a class-concept ''), whereas a complex whole can never 
be one of its own constituents. 

71. Class may be defined either extensionally or intensionally. 
That is to say, we may define the kind of object which is a class, or the 
kind of concept which denotes a class : this is the precise meaning of 
the opposition of extension and intension in this connection. But 
although the general notion can be defined in this two -fold manner, 
particular classes, except when they happen to be finite, can only be 
defined intensionally, ix. as the objects denoted by such and such con- 
cepts. I believe this distinction to be purely psychological : logically, 
the extensional definition appears to be equally applicable to infinite 
classes, but practically, if we were to attempt i,t. Death would cut short 
our laudable endeavour before it had attained its goal. Logically, 
therefore, extension and intension seem to be on a par. I will begin 
with the extensional view. 

When a class is regarded as defined by the enumeration of its terms, 
it is more naturally called a collection. I shall for the moment adopt 
this name, as it will not prejudge the question whether the objects 
denoted by it are truly classes or not. By a collection I mean what is 
conveyed by "A and .B'' or "A and B and C,'' or any other enumeration 
of definite terms. The collection is defined by the actual mention of 
the terms, and the terms are connected by and. It would seem that 
and represents a fundamental way of combining terms, and that just 
this way of combination is essential if anything is to result of which a 
number other than 1 can be asserted. Collections do not presuppose 
numbers, since they result simply from the terms together with and: 
they could only presuppose numbers in the particular case where the 
terms of the collection themselves presupposed numbers. There is a 
grammatical difficulty which, since no method exists of avoiding it, 
must be pointed out and allowed for. A collection, grammatically, is 

"^ A plurality of terms is not the logical subject when a number is asserted of it : 
such propositions have not one subject^ but many subjects. See end of § 74. 



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70 The Indeji7iables of Mathematics [chap, vi 

singular, whereas A and B^ A and B and C, etc. are essentially plural. 
This grammatical difficulty arises from the logical fact (to be discussed 
presently) that whatever is many in general forms a whole which is 
one ; it is, therefore, not removable by a better choice of technical 
terms. 

The notion oi and was brought into prominence by Bolzano*. In 
order to understand what infinity is, he says, " we must go back to one 
of the simplest conceptions of our understanding, in order to reach an 
agreement concerning the word that we are to use to denote it. This is 
the conception which underlies the conjunction and^ which, however, if 
it is to stand out as clearly as is required, in many cases, both by the 
purposes of mathematics and by those of philosophy, I believe to be best 
expressed by the words : ' A system (Inbegriff') of certain things,' or 
' a whole consisting of certain parts.' But we must add that every 
arbitrary object A can be combined in a system with any others 
B, Cy D^ ,.., or (speaking still more correctly) already forms a system 
by itself f, of which some more or less important truth can be enunciated, 
provided only that each of the presentations A^ jB, C, Z>, ... in fact 
represents a different object, or in so far as none of the propositions 
'A is the same as ^,' 'A is the same as C,' 'A is the same as D,' etc., 
is true. For if e.g, A is the same as B^ then it is certainly unreasonable 
to speak of a system of the things A and jB.'' 

The above passage, good as it is, neglects several distinctions which 
we have found necessary. First and foremost, it does not distinguish 
the many from the whole which they form. Secondly, it does not appear 
to observe that the method of enumeration is not practically applicable 
to infinite systems. Thirdly, and this is connected with the second point, 
it does not make any mention of intensional definition nor of the notion 
of a class. What we have to consider is the difference, if any, of a class 
from a collection on the one hand, and from the whole formed of the 
collection on the other. But let us first examine further the notion 
of and. 

Anything of which a finite number other than or 1 can be asserted 
would be commonly said to be many, and many, it might be said, are 
always of the form "A and B and C and ...." Here A, B^ C, ... are 
each one and are all different. To say that A is one seems to amount 
to much the same as to say that A is not of the form " Ai and A 2 and 
As and ....'' To say that A, B, C^ ... are all different seems to amount 
only to a condition as regards the symbols : it should be held that 
" A and A '' is meaningless, so that diversity is implied by and^ and need 
not be specially stated. 

A term A which is one may be regarded as a particular case of a 

* Pa^'adoocien des Unendlichen, Leipzig^ 1854 (2nd ed.^ Berlin^ 1889)^ § 3. 
+ i.e. the combination of A with B, G, D, ... already forms a system. 



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71] Classes 71 

collection, namely as a collection of one term. Thus every collection 
which is many presupposes many collections which are each one: A and 
B presupposes A and presupposes B, Conversely some collections of 
one term presuppose many, namely those which are complex: thus 
"A differs from jB'' is one, but presupposes A and difference and B. 
But there is not symmetry in this respect, for the ultimate presupposi- 
tions of anything are always simple terms. 

Every pair of terms, without exception, can be combined in the 
manner indicated by A and B^ and if neither A nor B be many, then 
A and B are two. A and B may be any conceivable entities, any 
possible objects of thought, they may be points or numbers or true or 
false propositions or events or people, in short anything that can be 
counted. A teaspoon and the number 3, or a chimaera and a four- 
dimensional space, are certainly two. Thus no restriction whatever is 
to be placed on A and B^ except that neither is to be many. It should 
be observed that A and B need not exist, but must, like anything that 
can be mentioned, have Being. The distinction of Being and existence 
is important, and is well illustrated by the process of counting. What 
can be counted must be something, and must certainly be^ though it 
need by no means be possessed of the further privilege of existence. 
Thus what we demand of the terms of our collection is merely that each 
should be an entity. 

The question may now be asked : What is meant by A and B ? 
Does this mean anything more than the juxtaposition of A with B ? 
That is, does it contain any element over and above that of A and that 
of B? Is and a separate concept, which occurs besides A^ B? To 
either answer there are objections. In the first place, and, we might 
suppose, cannot be a new concept, for if it were, it would have to be 
some kind of relation between A and B; A and B would then be a 
proposition, or at least a propositional concept, and would be one, not 
two. Moreover, if there are two concepts, there are two, and no third 
mediating concept seems necessary to make them two. Thus and would 
seem meaningless. But it is difficult to maintain this theory. To begin 
with, it seems rash to hold that any word is meaningless. When we use 
the word and, we do not seem to be uttering mere idle breath, but some 
idea seems to correspond to the word. Ag^in some kind of combination 
seems to be implied by the fact that A and B are two, which is not true 
of either separately. When we say "A and B are yellow,'" we can replace 
the proposition by " A is yellow '" and " B is yellow '' ; but this cannot 
be done for " A and B are two "*" ; on the contrary, A is one and B is on£. 
Thus it seems best to regard and as expressing a definite unique kind of 
combination, not a relation, and not combining A and B into a whole, 
which would be one. This unique kind of combination will in future be 
called addition of individuals. It is important to observe that it applies 
to terms, and only applies to numbers in consequence of their being 



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72 The Indejinables of Mathematics [chap, yi 

terms. Thus for the present, 1 and 2 are two, and 1 and 1 is 
meaningless. 

As regards what is meant by the combination indicated by and^ it is 
indistinguishable from what we before called a numerical conjunction. 
That is, A and B is what is denoted by the concept of a class of which 
A and B are the only members. If u be a class-concept of which the 
propositions "A is a u'''' "jB is a z^"' are true, but of which all other 
propositions of the same form are false, then " all z^'s " is the concept of 
a class whose only terms are A and B ; this concept denotes the terms 
A^ B combined in a certain way, and "A and B'''' are those terms com- 
bined in just that way. Thus " A and B " are the class, but are distinct 
from the class -concept and from the concept of the class. 

The notion of and^ however, does not enter into the meaning of a 
class, for a single term is a class, although it is not a numerical 
conjunction. If u be a class-concept, and only one proposition of the 
form ^' X is a z^'' be true, then "all z^'s'' is a concept denoting a single 
term, and this term is the class of which " all z^'s '' is a concept. Thus 
what seems essential to a class is not the notion of and^ but the being 
denoted by some concept of a class. This brings us to the intensional 
view of classes. 

72. We agreed in the preceding chapter that there are not 
different ways of denoting, but only different kinds of denoting concepts 
and correspondingly different kinds of denoted objects. We have 
discussed the kind of denoted object which constitutes a class ; we have 
now to consider the kind of denoting concept. 

The consideration of classes which results from denoting concepts 
is more general than the extensional consideration, and that in two 
respects. In the first place it allows, what the other practically 
excludes, the admission of infinite classes; in the second place it 
introduces the null concept of a class. But, before discussing these 
matters, there is a purely logical point of some importance to be 
examined. 

If u be a class-concept, is the concept " all z^'s " analyzable into two 
constituents, all and z/, or is it a new concept, defined by a certain 
relation to z^, and no more complex than u itself.^ We may observe, 
to begin with, that " all z/.'s " is synonymous with " z^'s," at least according 
to a very common use of the plural. Our question is, then, as to the 
meaning of the plural. The word all has certainly some definite 
meaning, but it seems highly doubtful whether it means more than 
the indication of a relation. " All men '' and " all numbers '' have in 
common the fact that they both have a certain relation to a class- 
concept, namely to man and number respectively. But it is very difficult 
to isolate any further element of all-ness which both share, unless we 
take as this element the mere fact that both are concepts of classes. 
It would seem, then, that "all z^'s'' is not validly analyzable into all 



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71-73] Classes 73 

and u^ and that language, in this case as in some others, is a misleading 
guide. The same remark will apply to every ^ any^ some, a, and the. 

It might perhaps be thought that a class ought to be considered, 
not merely as a numerical conjunction of terms, but as a numerical 
conjunction denoted by the concept of a class. This complication, 
however, would serve no useful purpose, except to preserve Peano^s 
distinction between a single term and the class whose only term it is — 
a distinction which is easy to grasp when the class is identified with the 
class-concept, but which is inadmissible in our view of classes. It is 
evident that a numerical conjunction considered as denoted is either 
the same entity as when not so considered, or else is a complex of 
denoting together with the object denoted; and the object denoted is 
plainly what we mean by a class. 

With regard to infinite classes, say the class of numbers, it is to be 
observed that the concept all numbers, though not itself infinitely 
complex, yet denotes an infinitely complex object. This is the inmost 
secret of our power to deal with infinity. An infinitely complex 
concept, though there may be such, can certainly not be manipulated 
by the human intelligence ; but infinite collections, owing to the notion 
of denoting, can be manipulated without introducing any concepts of 
infinite complexity. Throughout the discussions of infinity in later 
Parts of the present work, this remark should be borne in mind: if 
it is forgotten, there is an air of magic which causes the results obtained 
to seem doubtful. 

73. Great difficulties are associated with the null-class, and 
generally with the idea of nothing. It is plain that there is such a 
concept as nothing, and that in some sense nothing is something. In 
fact, the proposition " nothing is not nothing "" is undoubtedly capable 
of an interpretation which makes it true — a point which gives rise to 
the contradictions discussed in Plato's Sophist. In Symbolic Logic 
the null-class is the class which has no terms at all ; and symbolically 
it is quite necessary to introduce some such notion. We have to 
consider whether the contradictions which naturally arise can be 
avoided. 

It is necessary to realize, in the first place, that a concept may 
denote although it does not denote anything. This occurs when there 
are propositions in which the said concept occurs, and which are not 
about the said concept, but all such propositions are false. Or rather, 
the above is a first step towards the explanation of a denoting concept 
which denotes nothing. It is not, however, an adequate explanation. 
Consider, for example, the proposition "chimaeras are animals" or 
^' even primes other than S are numbers.'** These propositions appear 
to be true, and it would seem that they are not concerned with the 
denoting concepts, but with what these concepts denote ; yet that is 
impossible, for the concepts in question do not denote anything. 



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74 The Indejinables of Mathematics [chap, vi 

Symbolic Logic says that these concepts denote the null-class, and that 
the propositions in question assert that the null-class is contained in 
certain other classes. But with the strictly extensional view of classes 
propounded above, a class which has no terms fails to be anything at 
all : what is merely and solely a collection of terms cannot subsist when 
all the terms are removed. Thus we must either find a different 
interpretation of classes, or else find a method of dispensing with 
the null-class. 

The above imperfect definition of a concept which denotes, but 
does not denote anything, may be amended as follows. All denoting 
concepts, as we saw, are derived from class -concepts ; and a is a class - 
concept when " ^ is an c^ " is a prepositional function. The denoting 
concepts associated with a will not denote anything when and only 
when " 00 is an «" is false for all values of oo. This is a complete 
definition of a denoting concept which does not denote anything ; and 
in this case we shall say that a is a null class-concept, and that " all a\ "' 
is a null concept of a class. Thus for a system such as Peano's, in 
which what are called classes are really class-concepts, technical difficulties 
need not arise ; but for us a genuine logical problem remains. 

The proposition " chimaeras are animals '' may be easily interpreted 
by means of formal implication, as meaning "^ is a chimaera implies 
DC is an animal for all values of ^.'' But in dealing with classes we 
have been assuming that propositions containing all or any or every ^ 
though equivalent to formal implications, were yet distinct from them, 
and involved ideas requiring independent treatment. Now in the case 
of chimaeras, it is easy to substitute the pure intensional view, according 
to which what is really stated is a relation of predicates : in the case in 
question the adjective animal is part of the definition of the adjective 
chimerical (if we allow ourselves to use this word, contrary to usage, 
to denote the defining predicate of chimaeras). But here again it is 
fairly plain that we are dealing with a proposition which implies that 
chimaeras are animals, but is not the same proposition — indeed, in the 
present case, the implication is not even reciprocal. By a negation 
we can give a kind of extensional interpretation : nothing is denoted 
by a chimaera which is not denoted by an animal. But this is a very 
roundabout interpretation. On the whole, it seems most correct to 
reject the proposition altogether, while retaining the various other 
propositions that would be equivalent to it if there were chimaeras. 
By symbolic logicians, who have experienced the utility of the null- 
class, this will be felt as a reactionary view. But I am not at present 
discussing what should be done in the logical calculus, where the 
established practice appears to me the best, but what is the philo- 
sophical truth concerning the null-class. We shall say, then, that, 
of the bundle of normally equivalent interpretations' of logical symbolic 
formulae, the class of interpretations considered in the present chapter, 



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73] Classes 75 

which are dependent upon actual classes, fail where we are concerned 
with null class -concepts, on the ground that there is no actual null-class. 

We may now reconsider the proposition " nothing is not nothing "' — 
a proposition plainly true, and yet, unless carefully handled, a source of 
apparently hopeless antinomies. Nothing is a denoting concept, which 
denotes nothing. The concept which denotes is of course not nothing, 
i.e. it is not denoted by itself. The proposition which looks so para- 
doxical means no more than this : Nothings the denoting concept, is 
not nothing, i.e. is not what itself denotes. But it by no means follows 
from this that there is an actual null-class : only the null class-concept 
and the null concept of a class are to be admitted. 

But now a new difficulty has to be met. The equality of class- 
concepts, like all relations which are reflexive, symmetrical, and transitive, 
indicates an underlying identity, i.e. it indicates that every class -concept 
has to some term a relation which all equal class-concepts also have to 
that term — the term in question being different for different sets of 
equal class-concepts, but the same for the various members of a single 
set of equal class-concepts. Now for all class-concepts which are not 
null, this term is found in the corresponding class ; but where are we 
to find it for null class-concepts ? To this question several answers may 
be given, any of which may be adopted. For we now know what a 
class is, and we may therefore adopt as our term the class of all null 
class-concepts or of all null propositional functions. These are not null- 
classes, but genuine classes, and to either of them all null class -concepts 
have the same relation. If we then wish to have an entity analogous 
to what is elsewhere to be called a class, but corresponding to null 
class-concepts, we shall be forced, wherever it is^iecessary (as in counting 
classes) to introduce a term which is identical for equal class-concepts, 
to substitute everywhere the class of class-concepts equal to a given 
class-concept for the class corresponding to that class-concept. The 
class corresponding to the class -concept remains logically fundamental, 
but need not be actually employed in our symbolism. The null-class, 
in fact, is in some ways analogous to an irrational in Arithmetic : it 
cannot be interpreted on the same principles as other classes, and if 
we wish to give an analogous interpretation elsewhere, we must substitute 
for classes other more complicated entities — in the present case, certain 
correlated classes. The object of such a procedure will be mainly 
technical ; but failure to understand the procedure will lead to in- 
extricable difficulties in the interpretation of the symbolism. A very 
closely analogous procedure occurs constantly in Mathematics, for 
example with every generalization of number ; and so far as I know, 
no single case in which it occurs has been rightly interpreted either by 
philosophers or by mathematicians. So many instances will meet us 
in the course of the present work that it is unnecessary to linger longer 
over the point at present. Only one possible misunderstanding must 



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76 The Indefinables of Mathematics [chap, vi 

be guarded against. No vicious circle is involved in the above account 
of the null-class ; for the general notion of class is first laid down, is 
found to involve what is called existence, is then symbolically, not 
philosophically, replaced by the notion of a class of equal class-concepts, 
and is found, in this new form, to be applicable to what corresponds to 
null class -concepts, since what corresponds is now a class which is not 
null. Between classes simplicite?^ and classes of equal class -concepts 
there is a one-one correlation, which breaks down in the sole case of the 
class of null class-concepts, to which no null-class corresponds ; and this 
fact is the reason for the whole complication. 

74. A question which is very fundamental in the philosophy of 
Arithmetic must now be discussed in a more or less preliminary fashion. Is 
a class which has many terms to be regarded as itself one or many.? Taking 
the class as equivalent simply to the numerical conjunction "A and B 
and C and etc.,^' it seems plain that it is many ; yet it is quite necessary 
that we should be able to count classes as one each, and we do habitually 
speak of a class. Thus classes would seem to be one in one sense and 
many in another. 

There is a certain temptation to identify the class as many and the 
class as one, e.g., all men and the human race. Nevertheless, wherever 
a class consists of more than one term, it can be proved that no such 
identification is permissible. A concept of a class, if it denotes a class 
as one, is not the same as any concept of the class which it denotes. 
That is to say, classes of all rational animals., which denotes the human 
race as one term, is different from men., which denotes men, i.e. the 
human race as many. But if the human race were identical with men, 
it would follow that whatever denotes the one must denote the other, 
and the above difference would be impossible. We might be tempted 
to infer that Peano's distinction, between a term and a class of which 
the said term is the only member, must be maintained, at least when the 
term in question is a class*. But it is more correct, I think, to infer an 
ultimate distinction between a class as many and a class as one, to 
hold that the many are only many, and are not also one. The class as 
one may be identified with the whole composed of the terms of the class, 
i.e., in the case of men, the class as one will be the human race. 

But can we now avoid the contradiction always to be feared, 
where there is something that cannot be made a logical subject ? 
I do not myself see any way of eliciting a precise contradiction in this 
case. In the case of concepts, we were dealing with what was plainly 
one entity ; in the present case, we are dealing with a complex essentially 
capable of analysis into units. In such a proposition as "^ and B are 
two,'^ there is no logical subject: the assertion is not about A, nor 

^ This conclusion is actually drawn by Frege from an analogous argument : 
Archmfilr syst. Phil, i, p. 444. See Appendix. 



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73-76] Classes 77 

about B^ nor about the whole composed of both, but strictly and only 
about A and B, Thus it would seem that assertions are not necessarily 
about single subjects, but may be about many subjects ; and this removes 
the contradiction which arose, in the case of concepts, from the im- 
possibility of making assertions about them unless they were turned 
into subjects. This impossibility being here absent, the contradiction 
which was to be feared does not arise. 

75. We may ask, as suggested by the above discussion, what is to be 
said of the objects denoted by a man^ every man^ some man^ and any man. 
Are these objects one or many or neither ? Grammar treats them all as 
one. But to this view, the natural objection is, which one ? Certainly 
not Socrates, nor Plato, nor any other particular person. Can we 
conclude that no one is denoted ? As well might we conclude that 
every one is denoted, which in fact is true of the concept every man, 
I think one is denoted in every case, but in an impartial distributive 
manner. Any number is neither 1 nor % nor any other particular number, 
whence it is easy to conclude that any number is not any one number, 
a proposition at first sight contradictory, but really resulting from an 
ambiguity in any^ and more correctly expressed by " any number is not 
some one number.'** There are, however, puzzles in this subject which 
I do not yet know how to solve. 

A logical difficulty remains nn regard to the nature of the whole 
composed of all the terms of a class. Two propositions appear self- 
evident : (1) Two wholes composed of different terms must be different ; 
(^) A whole composed of one term only is that one term. It follows 
that the whole composed of a class considered as one term is that class 
considered as one term, and is therefore identical with the whole 
composed of the terms of the class ; but this result contradicts the 
first of our supposed self-evident principles. The answer in this case, 
however, is not difficult. The first of our principles is only universally 
true when all the terms composing our two wholes are simple. A given 
whole is capable, if it has more than two parts, of being analyzed in a 
plurality of ways ; and the resulting constituents, so long as analysis 
IS not pushed as far as possible, will be different for different ways of 
analyzing. This proves that different sets of constituents may constitute 
the same whole, and thus disposes of our difficulty. 

76. Something must be said as to the relation of a term to a class 
of which it is a member, and as to the various allied relations. One of 
the allied relations is to be called e, and is to be fundamental in Symbolic 
Logic. But it is to some extent optional which of them we take as 
symbolically fundamental. 

Logically, the fundamental relation is that of subject and predicate, 
expressed in " Socrates is human " — a relation which, as we saw in 
Chapter iv, is peculiar in that the relatum cannot be regarded as a term 



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78 The Indefinables of Mathematics [chap, yi 

in the proposition. The first relation that grows out of this is the one 
expressed by " Socrates has humanity,"" which is distinguished by the 
fact that here the relation is a term. Next comes "Socrates is a 
man." This proposition, considered as a relation between Socrates and 
the concept man^ is the one which Peano regards as fundamental ; and 
his e expresses the relation is a between Socrates and man. So long 
as we use class-concepts for classes in our symbolism, this practice is 
unobjectionable ; but if we give e this meaning, we must not assume 
that two symbols representing equal class -concepts both represent one 
and the same entity. We may go on to the relation between Socrates 
and the human race, i.e. between a term and its class considered as 
a whole ; this is expressed by " Socrates belongs to the human race." 
This relation might equally well be represented by e. It is plain that, 
since a class, except when it has one term, is essentially many, it cannot 
be as such represented by a single letter : hence in any possible Symbolic 
Logic the letters which do duty for classes cannot represent the classes 
as viany^ but must represent either class-concepts, or the wholes com- 
posed of classes, or some other allied single entities. And thus e cannot 
represent the relation of a term to its class as many ; for this would be 
a relation of one term to many terms, not a two-term relation such as 
we want. This relation might be expressed by " Socrates is one among 
men " ; but this, in any case, cannot be taken to be the meaning of e. 

77. A relation which, before Peano, was almost universally con- 
founded with 6, is the relation of inclusion between classes, as e.g. 
between men and mortals. This is a time-honoured relation, since 
it occurs in the traditional form of the syllogism : it has been a battle- 
ground between intension and extension, and has been so much dis- 
cussed that it is astonishing how much remains to be said about it. 
Empiricists hold that such propositions mean an actual enumeration 
of the terms of the contained class, with the assertion, in each case, 
of membership of the containing class. They must, it is to be in- 
ferred, regard it as doubtful whether all primes are integers, since they 
will scarcely have the face to say that they have examined all primes 
one by one. Their opponents have usually held, on the contrary, that 
what is meant is a relation of whole and part between the defining 
predicates, but turned in the opposite sense from the relation between 
the classes : i.e. the defining predicate of the larger class is part of that 
of the smaller. This view seems far more defensible than the other; 
and wherever such a relation does hold between the defining predicates, 
the relation of inclusion follows. But two objections may be made, 
first, that in some cases of inclusion there is no such relation between 
the defining predicates, and secondly, that in any case what is meant 
is a relation between the classes, not a relation of their defining 
predicates. The first point may be easily established by instances. 



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76-78] Classes 79 

The concept even prime does not contain as a constituent the concept 
integer between 1 and 10; the concept "Enghsh King whose head was 
cut off" does not contain the concept "people who died in 1649"; and 
so on through innumerable obvious cases. This might be met by saying 
that, though the relation of the defining predicates is not one of whole 
and part, it is one more or less analogous to implication, and is always 
what is really meant by propositions of inclusion. Such a view repre- 
sents, I think, what is said by the better advocates of intension, and 
I am not concerned to deny that a relation of the kind in question does 
always subsist between defining predicates of classes one of which is 
contained in the other. But the second of the above points remains 
valid as against any intensional interpretation. When we say that 
men are mortals, it is evident that we are saying something about men, 
not about the concept man or the predicate human. The question is, 
then, what exactly are we saying ? 

Peano held, in earlier editions of his Formulaire^ that what is 
asserted is the formal implication " ^ is a man implies ^ is a mortal." 
This is certainly implied, but I cannot persuade myself that it is the 
same proposition. For in this proposition, as we saw in Chapter iii, 
it is essential that x should take all values, and not only such as are 
men. But when we say " all men are mortals," it seems plain that we 
are only speaking of men, and not of all other imaginable terms. We 
may, if we wish for a genuine relation of classes, regard the assertion 
as one of whole and part between the two classes each considered as 
a single term. Or we may give a still more purely extensional form 
to our proposition, by making it mean : Every (or any) man is a mortal. 
This proposition raises very interesting questions in the theory of 
denoting : for it appears to assert an identity, yet it is plain that what 
is denoted by every man is different from what is denoted by a mortal. 
These questions, however, interesting as they are, cannot be pursued 
here. It is only necessary to realize clearly what are the various 
equivalent propositions involved where one class is included in another. 
The form most relevant to Mathematics is certainly the one with formal 
implication, which will receive a fresh discussion in the following 
chapter. 

Finally, we must remember that classes are to be derived, by means 
of the notion of such that^ from other sources than subject-predicate 
propositions and their equivalents. Any propositiorial function in 
which a fixed assertion is made of a variable term is to be regarded, 
as was explained in Chapter ii, as giving rise to a class of values 
satisfying it. This topic requires a discussion of assertions ; but one 
strange contradiction, which necessitates the care in discrimination 
aimed at in the present chapter, may be mentioned at once. 

78. Among predicates, most of the ordinary instances cannot be 



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80 The Indefinables of Mathematics [chap, yi 

predicated of themselves, though, by introducing negative predicates, 
it will be found that there are just as many instances of predicates which 
are predicable of themselves. One at least of these, namely predicability, 
or the property of being a predicate, is not negative : predicability, as 
is evident, is predicable, i.e. it is a predicate of itself. But the most 
common instances are negative : thus non-humanity is non-human, and 
so on. The predicates which are not predicable of themselves are, 
therefore, only a selection from among predicates, and it is natural to- 
suppose that they form a class having a defining predicate. But if so, 
let us examine whether this defining predicate belongs to the class or 
not. If it belongs to the class, it is not predicable of itself, for that is 
the characteristic property of the class. But if it is not predicable 
of itself, then it does not belong to the class whose defining predicate 
it is, which is contrary to the "hypothesis. On the other hand, if it 
does not belong to the class whose defining predicate it is, then it is not 
predicable of itself, i.e. it is one of those predicates that are not pre- 
dicable of themselves, and therefore it does belong to the class whose 
defining predicate it is — again contrary to the hypothesis. Hence from 
either hypothesis we can deduce its contradictory. I shall return to 
this contradiction in Chapter x ; for the present, I have introduced 
it merely as showing that no subtlety in distinguishing is likely to be 
excessive. 

79. To sum up the above somewhat lengthy discussion. A class, 
we agreed, is essentially to be interpreted in extension ; it is either 
a single term, or that kind of combination of terms which is indicated 
when terms are connected by the word and. But practically, though 
not theoretically, this purely extensional method can only be applied 
to finite classes. All classes, whether finite or infinite, can be obtained 
as the objects denoted by the plurals of class-concepts — men, numbers, 
points, etc. Starting with predicates, we distinguished two kinds of 
proposition, typified by " Socrates is human '' and " Socrates has 
humanity," of which the first uses human as predicate, the second 
as a term of a relation. These two classes of propositions, though 
very important logically, are not so relevant to Mathematics as their 
derivatives. Starting from human., we distinguished (1) the class-concept 
man^ which differs slightly, if at all, from human ; (S) the various 
denoting concepts all men^ every man^ any man^ a man and some man ; 
(8) the objects denoted by these concepts, of which the one denoted by 
all men was called the class as many^ so that all men (the concept) was 
called the concept of the class ; (4) the class as one, i.e. the human race. 
We had also a classification of propositions about Socrates, dependent 
upon the above distinctions, and approximately parallel with them : 
(1) " Socrates is-a man " is nearly, if not quite, identical with " Socrates 
has humanity '' ; (S) " Socrates is a-man "" expresses identity between 



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78, 79] Classes 81 

Socrates and one of the terms denoted by a man ; (8) " Socrates is one 
among men,^' a proposition which raises difficulties owing to the plurahty 
of men; (4) " Socrates belongs to the human race," which alone expresses 
a relation of an individual to its class, and, as the possibility of relation 
requires, takes the class as one, not as many. We agreed that the null- 
class, which has no terms, is a fiction, though there are null class-concepts„ 
It appeared throughout that, although any symbolic treatment must 
work largely with class -concepts and intension, classes and extension are 
logically more fundamental for the principles of Mathematics ; and this 
may be regarded as our main general conclusion in the present chapter. 



R. 



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CHAPTER VII. 



PROPOSITIONAL FUNCTIONS. 

80. In the preceding chapter an endeavour was made to indicate 
the kind of object that is to be called a class, and for purposes of 
discussion classes were considered as derived from subject-predicate 
propositions. This did not affect our view as to the notion of class 
itself; but if adhered to, it would greatly restrict the extension of 
the notion. It is often necessary to recognize as a class an object 
not defined by means of a subject-predicate proposition. The explana- 
tion of this necessity is to be sought in the theory of assertions and 
^uch that. 

The general notion of an assertion has been already explained in 
connection with formal implication. In the present chapter its scope 
and legitimacy are to be critically examined, and its connection with 
classes and such that is to be investigated. The subject is full of 
difficulties, and the doctrines which I intend to advocate are put forward 
with a very limited confidence in their truth. 

The notion of such that might be thought, at first sight, to be 
capable of definition ; Peano used, in fact, to define the notion by the 
proposition " the x'^ such that <x is an a are the class a^ Apart from 
further objections, to be noticed immediately, it is to be observed that 
the class as obtained from such that is the genuine class, taken in 
extension and as many, whereas the a m. " x is an a " is not the class, 
but the class-concept. Thus it is formally necessary, if Peano^s pro- 
cedure is to be permissible, that we should substitute for " ^'s such that 
so-and-so"' the genuine class-concept "x such that so-and-so,'' which 
may be regarded as obtained from the predicate ''such that so-a:nd-so" 
or rather, ''being an x such that so-and-so," the latter form being 
necessary because so-and-so is a propositional function containing x. 
But when this purely formal emendation has been made the point 
remains that such that must often be put before such propositions as 
wRa^ where J? is a given relation and a a given term. We cannot 
reduce this proposition to the form " x is an a ''''' without using such that ; 
for if we ask what cl must be, the answer is : a must be such that each 



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80, 81] Propositional Functions 83 

of its terms, and no other terms, have the relation R to a. To take 
examples from daily life : the children of Israel are a class defined by 
a certain relation to Israel, and the class can only be defined as the 
terms such that they have this relation. Such that is roughly equivalent 
to who or which^ and represents the general notion of satisfying a 
propositional function. But we may go further : given a class a, we 
cannot define, in terms of a, the class of propositions " ^ is an a '' for 
different values of x. It is plain that there is a relation which each 
of these propositions has to the x which occurs in it, and that the 
relation in question is determinate when a is given. Let us call the 
relation R. Then any entity which is a referent with respect to R 
is a proposition of the type '' x is an a^ But here the notion of 
such that is already employed. And the relation R itself can only be 
defined as the relation which holds between " x is an ^ " and x for all 
values of x^ and does not hold between any other pairs of terms. Here 
such that again appears. The point which is chiefly important in these 
remarks is the indefinability of propositional functions. When these 
have been admitted, the general notion of one- valued functions is easily 
defined. Every relation which is many-one, i.e. every relation for which 
a given referent has only one relatum, defines a function : the relatum 
is that function of the referent which is defined by the relation in 
question. But where the function is a proposition, the notion involved 
is presupposed in the symbolism, and cannot be defined by means of it 
without a vicious circle : for in the above general definition of a function 
propositional functions already occur. In the case of propositions of 
the type "x is an a^"* if we ask tvhat propositions are of this type, 
we can only answer " all propositions in which a term is said to be a'''' ; 
and here the notion to be defined reappears. 

81. Can the indefinable element involved in propositional func- 
tions be identified with assertion together with the notion of every 
proposition containing a given assertion, or an assertion made concerning 
every term ? The only alternative, so far as I can see, is to accept the 
general notion of a propositional function itself as indefinable, and for 
formal purposes this course is certainly the best ; but philosophically, 
the notion appears at first sight capable of analysis, and- we have to 
examine whether or not this appearance is deceptive. 

We saw in discussing verbs, in Chapter iv, that when a proposition 
is completely analyzed into its simple constituents, these constituents 
taken together do not reconstitute it. A less complete analysis of 
propositions into subject and assertion has also been considered; and 
this analysis does much less to destroy the proposition. A subject and 
an assertion, if simply juxtaposed, do not, it is true, constitute a 
proposition ; but as soon as the assertion is '^gfeually asserted of the 
subject, the proposition reappears. The assertion is everything that 
remains of the proposition when the subject is omitted : the verb 

6—2 



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84 The Indefinables of Mathematics [chap, vn 

remains an asserted verb, and is not turned into a verbal noun ; or at . 
any rate the verb retains that curious indefinable intricate relation to 
the other terms of the proposition which distinguishes a relating relation 
from the same relation abstractly considered. It is the scope and 
legitimacy of this notion of assertion which is now to be examined. 
Can every proposition be regarded as an assertion concerning any term 
occurring in it, or are limitations necessary as to the form of the 
proposition and the way in which the term enters into it ? 

In some simple cases, it is obvious that the analysis into subject 
and assertion is legitimate. In "Socrates is a man,^^ we can plainly 
distinguish Socrates and something that is asserted about him ; we 
should admit unhesitatingly that the same thing may be said about 
Plato or Aristotle. Thus we can consider a class of propositions 
containing this assertion, and this will be the class of which a typical 
number is represented by " ^ is a man.'' It is to be observed that the 
assertion must appear as assertion, not as term : thus " to be a man 
is to suffer" contains the same assertion, but used as term, and this 
proposition does not belong to the class considered. In the case of 
propositions asserting a fixed relation to a fixed term, the analysis 
seems equally undeniable. To be more than a yard long, for example, 
is a perfectly definite assertion, and we may consider the class of 
propositions in which this assertion is made, which will be represented 
by the propositional function " oo is more than a yard long.'' In such 
phrases as "snakes which are more than a yard long," the assertion 
appears very plainly; for it is here explicitly referred to a variable 
subject, not asserted of any one definite subject. Thus if i^ be a fixed 
relation and a a fixed term, . . . Ra is a perfectly definite assertion. 
(I place dots before the J?, to indicate the place where the subject 
must be inserted in order to make a proposition.) It may be doubted 
whether a relational proposition can be regarded as an assertion con- 
cerning the relatum. For my part, I hold that this can be done except 
in the case of subject-predicate propositions ; but this question is better 
postponed until we have discussed relations *. 

82. More difficult questions must now be considered. Is such 
a proposition as " Socrates is a man implies Socrates is a mortal," or 
"Socrates has a wife implies Socrates has a father," an assertion con- 
cerning Socrates or not ? It is quite certain that, if we replace Socrates 
by a variable, we obtain a propositional function ; in fact, the truth 
of this function for all values of the variable is what is asserted in the 
corresponding formal implication, which does not, as might be thought 
at first sight, assert a relation between two propositional functions. 
Now it was our intention, if possible, to explain propositional functions 
by means of assertions ; hence, if our intention can be carried out, the 

* See § 96. 



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81, 82] Propositional Functions 85 

above propositions must be assertions concerning Socrates. There is, 
however, a very great difficulty in so regarding them. An assertion was 
to be obtained from a proposition by simply omitting one of the terms 
occurring in the proposition. But when we omit Socrates, we obtain 
^' ... is a man implies ... is a mortal.'' In this formula it is essential 
that, in restoring the proposition, the same term should be substituted 
in the two places where dots indicate the necessity of a term. It does 
not matter what term we choose, but it must be identical in both places. 
Of this requisite, however, no trace whatever appears in the would-be 
assertion, and no trace can appear, since all mention of the term to be 
inserted is necessarily omitted. When an x is inserted to stand for 
the variable, the identity of the term to be inserted is indicated by the 
repetition of the letter oc ; but in the assertional form no such method is 
available. And yet, at first sight, it seems very hard to deny that the 
proposition in question tells us a fact about Socrates, and that the same 
fact is true about Plato or a plum-pudding or the number S. It is 
certainly undeniable that " Plato is a man implies Plato is a mortal " 
is, in some sense or other, the same function of Plato as our previous 
proposition is of Socrates. The natural interpretation of this statement 
would be that the one proposition has to Plato the same relation as the 
other has to Socrates. But this requires that we should regard the 
propositional function in question as definable by means of its relation 
to the variable. Such a view, however, requires a propositional function 
more complicated than the one we are considering. If we represent 
" ^ is a man implies <r is a mortal " by (^oc^ the view in question maintains 
that ^00 is the term having to oc the relation J?, where R is some definite 
relation. The formal statement of this view is as follows : For all values 
of oc and ?/, " y is identical with c/)^" is equivalent to "y has the relation 
R to ^." It is evident that this will not do as an explanation, since it 
has far greater complexity than what it was to explain. It would seem 
to follow that propositions may have a certain constancy of form, ex- 
pressed in the fact that they are instances of a given propositional 
function, without its being possible to analyze the propositions into a 
constant and a variable factor. Such a view is curious and difficult : 
constancy of form, in all other cases, is reducible to constancy of rela- 
tions, but the constancy involved here is presupposed in the notion 
of constancy of relation, and cannot therefore be explained in the 
usual way. 

The same conclusion, I think, will result from the case of two 
variables. The simplest instance of this case is ocRy^ where J? is a 
constant relation, while x and y are independently variable. It seems 
evident that this is a propositional function of two independent variables : 
there is no difficulty in the notion of the class of all propositions of the 
form xRy. This class is involved — or at least all those members of 
the class that are true are involved — in the notion of the classes of 



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86 The Indefinables of Mathematics [chap, yii 

referents and relata with respect to J?, and these classes are unhesita- 
tingly admitted in such words as parents and children, masters and 
servants, husbands and wives, and innumerable other instances from 
daily life, as also in logical notions such as premisses and conclusions, 
causes and effects, and so on. All such notions depend upon the class 
of propositions ty])ified by xRy^ where R is constant while w and y are 
variable. Yet it is very difficult to regard xRy as analyzable into the 
assertion R concerning x and ?/, for the very sufficient reason that this 
view destroys the sense of the relation, i.e. its direction from x to y^ 
leaving us with some assertion which is symmetrical with respect to 
X and 2/5 such as "the relation R holds between x and y.'''' Given a 
relation and its terms, in fact, two distinct propositions are possible. 
Thus if we take R itself to be an assertion, it becomes an ambiguous 
assertion : in supplying the terms, if we are to avoid ambiguity, we 
must decide which is referent and which relatum. We may quite 
legitimately regard . . .Ry as an assertion, as was explained before ; but 
here y has become constant. We may then go on to vary y^ considering 
the class of assertions ...Ry for different values of «/; but this process 
does not seem to be identical with that which is indicated by the 
independent variability of x and y in the propositional function xRy. 
Moreover, the suggested process requires the variation of an element 
in an assertion, namely of y in ...Ry^ and this is in itself a new and 
difficult notion. 

A curious point arises, in this connection, from the consideration, 
often essential in actual Mathematics, of a relation of a term to itself. 
Consider the propositional function xRx^ where J? is a constant relation. 
Such functions are required, in considering, e.g.^ the class of suicides or 
of self-made men; or again, in considering the values of the variable 
for which it is equal to a certain function of itself, which may often be 
necessary in ordinary Mathematics. It seems exceedingly evident, in 
this case, that the proposition contains an element which is lost when 
it is analyzed into a term x and an assertion R. Thus here again, the 
propositional function must be admitted as fundamental. 

83. A difficult point arises as to the variation of the concept in a 
proposition. Consider, for example, all propositions of the type aJ?6, 
where a and h are fixed terms, and i? is a variable relation. There 
seems no reason to doubt that the class-concept " relation between a 
and h '' is legitimate, and that there is a corresponding class ; but this 
requires the admission of such propositional functions as aRh^ which, 
moreover, are frequently required in actual Mathematics, as, for example, 
in counting the number of many-one relations whose referents and relata 
are given classes. But if our variable is to have, as we normally 
require, an unrestricted field, it is necessary to substitute the pro- 
positional function " i? is a relation implies aRh.'''' In this proposition 
the implication involved is material, not formal. If the implication were 



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82, 83] Propositional Functions 87 

formal, the proposition would not be a function of J?, but would be 
equivalent to the (necessarily false) proposition : "AH relations hold 
between a and &." Generally we have some such proposition as " aRb 
implies (/> {R) provided i? is a relation," and we wish to turn this into a 
formal implication. If (^ (R) is a proposition for all values of i?, our 
object is effected by substituting "If 'i^ is a relation' imphes ' aRh^^ 
then (^ {R)y Here R can take all values *, and the if and then is a formal 
implication, while the implies is a material implication. If (^ (R) is not 
a propositional function, but is a proposition only when R satisfies '^{R)-, 
where '\\r (R) is a propositional function implied by '^ J? is a relation '' for 
all values of R, then our formal implication can be put in the form " If 
' i^ is a relation ' implies aRb, then, for all values of J?, yfr (R) implies 
(j) (R),'' where both the subordinate implications are material. As regards 
the material implication "' R is a relation ' implies aRb^'' this is always 
a proposition, whereas aRb is only a proposition when J? is a relation. 
The new propositional function will only be true when J? is a relation 
which does hold between a and b : when R is not a relation, the ante- 
cedent is false and the consequent is not a proposition, so that the 
implication is false ; when J? is a relation which does not hold between 
a and b^ the antecedent is true and the consequent false, so that again 
the implication is false ; only when both are true is the implication true. 
Thus in defining the class of relations holding between a and b, the 
formally correct course is to define them as the values satisfying "R 
is a i^elation implies aRb "" — an implication which, th^us^h it contains a 
variable, is not formal, but m-cerial, being satisfied by some only of the 
possible values of R, Fhe variable R in it is, in Peano's language, real 
and not apparent. 

The general principle involved is : If c/)^ is only a proposition for 
some values of ^, then " ' (f)a^ implies (j)a^ ' implies (f)^ '' is a proposition 
for all values of ^, and is true when and only when (j)X is true. (The 
implications involved are both material.) In some cases, "^^ implies 4>a^'"' 
will be equivalent to some simpler propositional function ^|ra^ (such as "R is 
a relation " in the above instance), which may then be substituted for itf. 

Such a propositional function as "i? is a relation imphes aRb'^ 
appears even less capable than previous instances of analysis into R and 
an assertion about J?, since we should have to assign a meaning to" a.. .5," 
where the blank space may be filled by anything, not necessarily by a 
relation. There is here, however, a suggestion of an entity which has 
not yet been considered, namely the couple with sense. It may be 
doubted whether there is any such entity, and yet such phrases as 

* It is necessary to assign some meaning (other than a proposition) to aRb when 
B is not a relation. 

t A propositional function,, though for every value of the variable it is true or 
falseyis not itself true or false^ being what is denoted by ^'^any proposition of the 
type in question/' which is not itself a proposition. 



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88 The Indefinables of Mathematics [chap, vii 

"R is a relation holding from a to 6 '' seem to show that its rejection 
would lead to paradoxes. This point, however, belongs to the theory 
of relations, and will be resumed in Chapter ix (§ 98). 

From what has been said, it appears that propositional functions 
must be accepted as ultimate data. It follows that formal implication 
and the inclusion of classes cannot be generally explained by means of a 
relation between assertions, although, where a propositional function 
asserts a fixed relation to a fixed term, the analysis into subject and 
assertion is legitimate and not unimportant. 

84. It only remains to say a few words concerning the derivation 
of classes from propositional functions. When we consider the ^'s such 
that <^^, where <^cc is a propositional function, we are introducing a 
notion of which, in the calculus of propositions, only a very shadowy use 
is made — I mean the notion of truth. We are considering, among 
all the propositions of the type ^^, those that are true: the corre- 
sponding values of oo give the class defined by the function ^oc. It must 
be held, I think, that every propositional function which is not null 
defines a class, which is denoted by "aP^ such that <^^.'' There is thus 
always a' concept of the class, and the class-concept corresponding will 
be the singular, " x such that ^x.'''' But it may be doubted — indeed the 
contradiction with which I ended the preceding chapter gives reason for 
doubting — whether there is always a defining predicate of such classes. 
Apart from the contradiction in question, this point might appear to be 
merely verbal: " being sin oj such that ^ r," it might be said, may always 
b^ takCxx to be a predicate. But in view of our contradiction, all 
remarks on this subject must be viewed with ca Uoii. This subject, 
however, will be resumed in Chapter x. 

85. It is to be observed that, according to the theory of pro- 
positional functions here advocated, the </> in (^oc is not a separate and 
distinguishable entity : it lives in the propositions of the form (/)<r, and 
cannot survive analysis. I am highly doubtful whether such a view does 
not lead to a contradiction, but it appears to be forced upon us, and it 
has the merit of enabling us to avoid a contradiction arising from the 
opposite view. If were a distinguishable entity, there would be a 
proposition asserting (/> of itself, which we may denote by (/) ((^) ; there 
would also be a proposition not-(/> ((^), denying </> (0). In this proposi- 
tion we may regard (j> as variable ; we thus obtain a propositional 
function. The question arises: Can the assertion in this propositional 
function be asserted of itself.? The assertion is non-assertibility of self, 
hence if it can be asserted of itself, it cannot, and if it cannot, it can. 
This contradiction is avoided by the recognition that the functional 
part of a propositional function is not an independent entity. As the 
contradiction in question is closely analogous to the other, concerning 
predicates not predicable of themselves, we may hope that a similar 
solution will apply there also. 



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CHAPTER YIII. 



THE VARIABLE. 



86. The discussions of the preceding chapter elicited the funda- 
mental nature of the variable ; no apparatus of assertions enables us to 
dispense with the consideration of the varying of one or more elements 
in a proposition while the other elements remain unchanged. The 
variable is perhaps the most distinctively mathematical of all notions ; 
it is certainly also one of the most difficult to understand. The attempt, 
if not the deed, belongs to the present chapter. 

The theory as to the nature of the variable, which results from our 
previous discussions, is in outline the following. When a given term 
occurs as term in a proposition, that term may be replaced by any other 
while the remaining terms are unchanged. The class of propositions 
so obtained have what may be called constancy of form, and this con- 
stancy of form must be taken as a primitive idea. The notion of a class 
of propositions of constant form is more fundamental than the general 
notion of class^ for the latter can be defined in terms of the former, 
but not the former in terms of the latter. Taking any term, a certain 
member of any class of propositions of constant form will contain that 
term. Thus ^, the variable, is what is denoted by any term^ and <^^, 
the propositional function, is what is denoted by the proposition of the 
form (f> in which x occurs. We may say that w is the oo is any ^oo^ where 
<^x denotes the class of propositions resulting from different values of x. 
Thus in addition to propositional functions, the notions of any and of 
denoting are presupposed in the notion of the variable. This theory, 
which, I admit, is full of difficulties, is the least objectionable that I 
have been able to imagine. I shall now set it forth more in detail. 

87. Let us observe, to begin with, that the explicit mention of 
any^ some^ etc., need not occur in Mathematics : formal implication will 
express all that is required. Let us recur to an instance already dis- 
cussed in connection with denoting, where a is a class and h a class 
of classes. We have 

" Any a belongs to any b "" is equivalent to ^^ ^^ is an a' implies that 
^ w is a V implies 'oo is a z/' " ; 



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90 The Indejinables of Mathematics [char yiii 

" Any a belongs to a &" is equivalent to "'x is an a' implies Hhere 
is a &5 say u^ such that x is a z^^^''*; 

"Any ^ belongs to some &*" is equivalent to "there is a &, say u^ such 
that ^«r is an a' implies '^ is a u^''''; 
and so on for the remaining relations considered in Chapter v. The 
question arises : How far do these equivalences constitute definitions of 
any^ a^ some^ and how far are these notions involved in the symbolism 
itself? 

The variable is, from the formal standpoint, the characteristic notion 
of Mathematics. Moreover it is the method of stating general theorems, 
which always mean something different from the intensional propositions 
to which such logicians as Mr Bradley endeavour to reduce them. That 
the meaning of an assertion about all men or any man is different from 
the meaning of an equivalent assertion about the concept man^ appears 
to me, I must confess, to be a self-evident truth— as evident as the fact 
that propositions about John are not about the name John. This point, 
therefore, I shall not argue further. That the variable characterizes 
Mathematics will be generally admitted, though it is not generally 
perceived to be present in elementary Arithmetic. Elementary Arith- 
metic, as taught to children, is characterized by the fact that the numbers 
occurring in it are constants ; the answer to any schoolboy's sum is 
obtainable without propositions concerning any number. But the fact 
that this is the case can only be proved by the help of propositions 
about any number, and thus we are led from schoolboy's Arithmetic to 
the Arithmetic which uses letters for numbers and proves general 
theorems. How very different this subject is from childhood's enemy may 
be seen at once in such works as those of Dedekindf and StolzJ. Now 
the difference consists simply in this, that our numbers have now become 
variables instead of being constants. We now prove theorems concern- 
ing n, not concerning 8 or 4 or any other particular number. Thus it is 
absolutely essential to any theory of Mathematics to understand the 
nature of the variable. 

Originally, no doubt, the variable was conceived dynamically, as 
something which changed with the lapse of time, or, as is said, as some- 
thing which successively assumed all values of a certain class. This 
view cannot be too soon dismissed. If a theorem is proved concerning 
7Z, it must not be supposed that n is a kind of arithmetical Proteus, 
which is 1 on Sundays and 2 on Mondays, and so on. Nor must it be 
supposed that n simulttoeously assumes all its values. If n stands for 
any integer, we cannot say that 7^ is 1, nor yet that it is 2, nor yet that 



■^ Here ^^ there is a cj" where c is any class^ is defined as equivalent to ^^If p 
implies p, and ^ a? is a c ' implies p for all values of or, then p is true." 
+ Was sind und was sollen die Zahlen ? Brunswick^ 1893. 
J AUgemeine Arithmetik, Leipzig^ 1885. 



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87-89] The Variable 91 

it is any other particular number. In fact, n just denotes any number, 
and this is something quite distinct from each and all of the numbers. 
It is not true that 1 is any number, though it is true that whatever 
holds of any number holds of 1. The variable, in short, requires the 
indefinable notion of any which was explained in Chapter v. 

88. We may distinguish what may be called the true or formal 
variable from the restricted variable. Any term is a concept denoting 
the true variable ; if u be a class not containing all terms, any u denotes 
a restricted variable. The terms included in the object denoted by the 
defining concept of a variable are called the values of the variable : thus 
every value of a variable is a constant. There is a certain difficulty 
about such propositions as " any number is a number.^^ Interpreted by 
formal implication, they offer no difficulty, for they assert merely that 
the propositional function " ^ is a number implies <a7 is a number '' holds 
for all values of oc. But if "any number''^ be taken to be a definite 
object, it is plain that it is not identical with 1 or 2 or 3 or any number 
that may be mentioned. Yet these are all the numbers there are, so 
that " any number " cannot be a number at all. The fact is that the 
concept "any number'' does denote one number, but not a particular 
one. This is just the distinctive point about a7iy^ that it denotes a term 
of a class, but in an impartial distributive manner, with no preference 
for one term over another. Thus although sc is a number, and no one 
number is ^, yet there is here no contradiction, so soon as it is recognized 
that X is not one definite term. 

The notion of the restricted variable can be avoided, except in regard 
to propositional functions, by the introduction of a suitable hypothesis, 
namely the hypothesis expressing the restriction itself. But in- respect 
of propositional functions this is not possible. The x in <^Xj where (^x 
is a propositional function, is an unrestricted variable ; but the (^x itself 
is restricted to the class which we may call (/>. (It is to be remembered 
that the class is here fundamental, for we found it impossible, without a 
vicious circle, to discover any common characteristic by which the class 
could be defined, since the statement of any common characteristic is 
itself a propositional function.) By making our x always an unrestricted 
variable, we can speak of the variable, which is conceptually identical in 
Logic, Arithmetic, Geometry, and all other formal subjects. The terms 
dealt with are always all terms ; only the complex concepts that occur 
distinguish the various branches of Mathematics. 

89. We may now return to the apparent definability of any^ some^ 
and a, in terms of formal implication. Let a and Z? be class-concepts, 
and consider the proposition " any a is a 6." This is to be interpreted 
as meaning " ^ is an a implies x is a h^ It is plain that, to begin with, 
the two propositions do not mean the same thing: for any 6^ is a concept 
denoting only a\^ whereas in the formal implication x need not be an a. 
But we might, in Mathematics, dispense altogether with " any « is a &,*" 



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92 The Indefinables of Mathematics [chap, yiii 

and content ourselves with the formal implication : this is, in fact, 
symbolically the best course. The question to be examined, therefore, 
is : How far, if at all, do any and some and a enter into the formal 
implication ? (The fact that the indefinite article appears in " oo is 
an a " and " ^ is a 5 '' is irrelevant, for these are merely taken as typical 
propositional functions.) We have, to begin with, a class of true 
propositions, each asserting of some constant term that if it is an a it is 
a h. We then consider the restricted variable, " any proposition of this 
class." We assert the truth of any term included among the values of 
this restricted variable. But in order to obtain the suggested fornmla, 
it is necessary to transfer the variability from the proposition as a whole 
to its variable term. In this way we obtain " .27 is an ^ implies w is Z?." 
But the genesis remains essential, for we are not here expressing a 
relation of two propositional functions " x is an a " and " ^ is a &." If 
this were expressed, we should not require the same x both times. Only 
one propositional function is involved, namely the whole formula. Each 
proposition of the class expresses a relation of one term of the pro- 
positional function " ^ is an <2 '^ to one of " ^ is a Z? '"' ; and we may say, 
if we choose, that the whole formula expresses a relation of any term of 
"x is an a" to some term of "x is a b^ We do not so much have 
an implication containing a variable as a variable implication. Or 
again, we may say that the first x is any term, but the second is some 
term, namely the first x. We have a class of implications not containing 
variables, and we consider any member of this class. If any member 
is true, the fact is indicated by introducing a typical implication con- 
taining a variable. This typical implication is what is called a formal 
implication: it is any member of a class of material implications. Thus 
it would seem that any is presupposed in mathematical formalism, but 
that some and a may be legitimately replaced by their equivalents in 
terms of formal implications. 

90. Although soine may be replaced by its equivalent in terms of 
any^ it is plain that this does not give the meaning of some. There is, 
in fact, a kind of duality of any and some : given a certain propositional 
function, if all terms belonging to the propositional function are asserted, 
we have any^ while if one at least is asserted (which gives what is called 
an existence-theorem), we get some. The proposition ^x asserted with- 
out comment, as in "-z* is a man implies .r is a mortal," is to be taken 
to mean that <^^ is true for all values of x (or for a,ny value), but it 
might equally well have been taken to mean that c^x is true for some 
value of X. In this way we might construct a calculus with two kinds 
of variable, the conjunctive and the disjunctive, in which the latter 
would occur wherever an existence-theorem was to be stated. But this 
method does not appear to possess any practical advantages. 

91. It is to be observed that what is fundamental is not particular 
propositional functions, but the class-concept propositional function, A 



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89-93] The Variable 93 

prepositional function is the class of all propositions which arise from 
the variation of a single term, but this is not to be considered as a 
definition, for reasons explained in the preceding chapter. 

92. From propositi onal functions all other classes can be derived 
by definition, with the help of the notion of such that. Given a pro- 
positional function </)ZC, the terms such that, when x is identified with 
any one of them, (f)X is true, are the class defined by ^x. This is the 
class as many, the class in extension. It is not to be assumed that every 
class so obtained has a defining predicate : this subject will be discussed 
afresh in Chapter x. But it must be assumed, I think, that a class in 
extension is defined by any prepositional function, and in particular 
that all terms form a class, since many prepositional functions {e.g. 
all formal implications) are true of all terms. Here, as with formal 
implications, it is necessary that the whole propositi onal function whose 
truth defines the class should be kept intact, and not, even where this 
is possible for every value of x^ divided into separate prepositional 
functions. For example, if a and h be two classes, defined by ^x and y\rx 
respectively, their common part is defined by the product i^x . a/t^, where 
the product has to be made for every value of x^ and then x varied 
afterwards. If this is not done, we do not necessarily have the same 
X in ^x and 'y\rx. Thus we do not multiply prepositional functions, but 
propositions : the new prepositional function is the class of products 
of corresponding propositions belonging to the previous functions, and 
is by no means the product of ^x and -v/r^. It is only in virtue of 
a definition that the logical product of the classes defined by <^,27 and '\\rx 
is the class defined by (\>x . -^x. And wherever a proposition containing 
an apparent variable is asserted, what is asserted is the truth, for all 
values of the variable or variables, of the prepositional function corre- 
sponding to the whole proposition, and is never a relation of prepositional 
functions. 

93. It appears from the above discussion that the variable is a 
very complicated logical entity, by no means easy to analyze correctly. 
The following appears to be as nearly correct as any analysis I can make. 
Given any proposition (not a prepositional function), let a be one of 
its terms, and let us call the proposition <^ {a). Then in virtue of the 
primitive idea of a prepositional function, if x be any term, we can 
consider the preposition ^ (<r), which arises from the substitution of x 
in place of a. We thus arrive at the class of all prepositions ^ {x). 
If all are true, <^ {x) is asserted simply : ^ {x) may then be called a 

formal truth. In a formal implication, (/> ix\for every value of x^ states 
an implication, and the assertion of ^ {pc) is the assertion of a class of 
implications, net of a single implication. If (x) is sometimes true, 
the values of x which make it true form a class, which is the class defined 
by <^ ix) : the class is said to exist in this case. If ^ (x) is false for all 
values of x, the class defined by (f> (x) is said net to exist, and as a 



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94 The Indefinables of Mathematics [chap, yiii 

matter of fact, as we saw in Chapter vi, there is no such class, if classes 
are taken in extension. . Thus x is, in some sense, the object denoted by 
any term ; yet this can hardly be strictly maintained, for different 
variables may occur in a proposition, yet the object denoted by any 
terrn^ one would suppose, is unique. This, however, elicits a new point 
in the theory of denoting, namely that aiiy term does not denote, 
properly speaking, an assemblage of terms, but denotes one term, only 
not one particular definite term. Thus any term may denote different 
terms in different places. We may say : any term has some relation to 
any term ; and this is quite a different proposition from : any term has 
some relation to itself. Thus variables have a kind of individuality. 
This arises, as I have tried to show, from propositional functions. 
When a propositional function has two variables, it must be regarded 
as obtained by successive steps. If the propositional function cf) {x^ y) 
is to be asserted for all values of x and «/, we must consider the assertion, 
for all values of «/, of the propositional function ch (a, y\ where a is 
a constant. This does not involve «/, and may be represented by -v/r (a). 
We then vary a, and assert -^(x) for all values of x. The process is 
analogous to double integration ; and it is necessary to prove formally 
that the order in which the variations are made makes no difference 
to the result. The individuality of variables appears to be thus ex- 
plained. A variable is not any term simply, but any term as entering 
into a propositional function. We may say, if (jb^ be a propositional 
function, that x is tlie term in any proposition of the class of proposi- 
tions whose type is ^x. It thus appears that, as regards propositional 
functions, the notions of class, of denoting, and of any^ are fundamental, 
being presupposed in the symbolism employed. With this conclusion, 
the analysis of formal implication, which has been one of the principal 
problems of Part I, is carried as far as I am able to carry it. May 
some reader succeed in rendering it more complete, and in answering the 
many questions which I have had to leave unanswered. 



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CHAPTER IX. 



RELATIONS. 

94. Next after subject-predicate propositions come two types of 
propositions which appear equally simple. These are the propositions 
in which a relation is asserted between two terms, and those in which 
two terms are said to be two. The latter class of propositions will be 
considered hereafter; the former must be considered at once. It has 
often been held that every proposition can be reduced to one of the 
subject-predicate type, but this view we shall, throughout the present 
work, find abundant reason for rejecting. It might be held, however, 
that all propositions not of the subject-predicate type, and not asserting 
numbers, could be reduced to propositions containing two terms and 
a relation. This opinion would be more difficult to refute, but this too, 
we shall find, has no good grounds in its favour*. We may therefore 
allow that there are relations having more than two terms ; but as these 
are more complex, it will be well to consider first such as have two 
terms only. 

A relation between two terms is a concept which occurs in a 
proposition in which there are two terms not occurring as concepts f, 
and in which the interchange of the two terms gives a different pro- 
position. This last mark is required to distinguish a relational 
proposition from one of the type " a and h are two,'' which is identical 
with " h and a are two." A relational proposition may be symbolized 
by aRh^ where R is the relation and a and h are the terms ; and aRb 
will then always, provided a and h are not identical, denote a different 
proposition from hRa. That is to say, it is characteristic of a relation 
of two terms that it proceeds, so to speak, yroT/z one to the other. This 
is what may be called the sense of the relation, and is, as we shall find, 
the source of order and series. It must be held as an axiom that aRh 
implies and is implied by a relational proposition hR'a^ in which the 

^ See inf., Part IV, Chap, xxv, § 200. 

t This description^ as we saw above (§ 48), excludes the pseudo-relation of subject 
to predicate. 



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96 The Indefinables of Mathematics [chap, ix 

relation R' proceeds from h to a^ and may or may not be the same 
relation as R. But even when aRh implies and is implied by hRa^ 
it must be strictly maintained that these are different propositions. 
We may distinguish the term from which the relation proceeds as the 
referent^ and the term to which it proceeds as the relatum. The sense 
of a relation is a fundamental notion, which is not capable of definition. 
The relation which holds between h and a whenever R holds between 
a and h will be called the converse of R^ and will be denoted (following 
Schroder) by R. The relation oi R to R is the relation of oppositeness, 
or difference of sense ; and this must not be defined (as would seem at 
first sight legitimate) by the above mutual implication in any single 
case, but only by the fact of its holding for all cases in which the given 
relation occurs. The grounds for this view are derived from certain 
propositions in which terms are related to themselves not-symmetrically, 
i.e. by a relation whose converse is not identical with itself. These 
propositions must now be examined. 

95. There is a certain temptation to affirm that no term can be 
related to itself ; and there is a still stronger temptation to affirm that, 
if a term can be related to itself, the relation must be symmetrical, 
i.e. identical with its converse. But both these temptations must be 
resisted. In the first place, if no term were related to itself, we should 
never be able to assert self -identity, since this is plainly a relation. 
But since there is such a notion as identity, and since it seems undeniable 
that every term is identical with itself, we must allow that a term may 
be related to itself. Identity, however, is still a symmetrical relation, 
and may be admitted without any great qualms. The matter becomes 
far worse when we have to admit not -symmetrical relations of terms 
to themselves. Nevertheless the following propositions seem undeniable ; 
Being is, or has being ; 1 is one, or has unity ; concept is conceptual : 
term is a term ; class-concept is a class-concept. All these are of one 
of the three equivalent types which we distinguished at the beginning of 
Chapter v, which may be called respectively subject-predicate proposi- 
tions, propositions asserting the relation of predication, and propositions 
asserting membership of a class. What we have to consider is, then, 
the fact that a predicate may be predicable of itself. It is necessary, for 
our present purpose, to take our propositions in the second form (Socrates 
has humanity), since the subject-predicate form is not in the above sense 
relational. We may take, as the type of such propositions, " unity has 
unity." Now it is certainly undeniable that the relation of predication 
is asymmetrical, since subjects cannot in general be predicated of their 
predicates. Thus "unity has unity'' asserts one relation of unity to 
itself, and implies another, namely the converse relation : unity has 
to itself both the relation of subject to predicate, and the relation of 
predicate to subject. Now if the referent and the relatum are identical, 
it is plain that the relatum has to the referent the same relation as the 



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94-96] Belations 97 

referent has to the relatum. Hence if the converse of a relation in 
a particular case were defined by mutual implication in that particular 
case, it would appear that, in the present case, our relation has two 
converses, since two different relations of relatum to referent are implied 
by ^' unity has unity."" We must therefore define the converse of a 
relation by the fact that aRh implies and is implied by hRa lohatevei^ 
a and b may be, and whether or not the relation R holds between them. 
That is to say, a and h are here essentially variables, and if we give 
them any constant value, we may find that aRh implies and is implied 
by hR'a^ where R' is some relation other than R. 

Thus three points must be noted with regard to relations of two 
terms : (1) they all have sense, so that, provided a and h are not 
identical, we can distinguish aRh from hRa ; (2) they all have a 

converse, ix, a relation R such that aRh implies and is implied by 
hRa^ whatever a and h may be ; (3) some relations hold between a 
term and itself, and such relations are not necessarily symmetrical, 
i.e. there may be two different relations, which are each other^s con- 
verses, and which both hold between a term and itself. 

96. For the general theory of relations, especially in its mathe- 
matical developments, certain axioms relating classes and relations are 
of great importance. It is to be held that to have a given relation to a 
given term is a pi^edicate, so that all terms having this relation to this 
term form a class. It is to be held further that to have a given relation 
at all is a predicate, so that all referents with respect to a given relation 
form a class. It follows, by considering the converse relation, that all 
relata also form a class. These two classes I shall call respectively the 
domain and the converse domain of the relation ; the logical sum of the 
two I shall call the Jield of the relation. 

The axiom that all referents with respect to a given relation form a 
class seems, however, to require some limitation, and that on account of 
the contradiction mentioned at the end of Chapter vi. This contra- 
diction may be stated as follows. We saw that some predicates can be 
predicated of themselves. Consider now those of which this is not the 
case. These are the referents (and also the relata) in what seems like 
a complex relation, namely the combination of non-predicability with 
identity. But there is no predicate which attaches to all of them and 
to no other terms. For this predicate will either be predicable or not 
predicable of itself. If it is predicable of itself, it is one of those 
referents by relation to which it was defined, and therefore, in virtue 
of their definition, it is not predicable of itself. Conversely, if it is not 
predicable of itself, then again it is one of the said referents, of all of 
which (by hypothesis) it is predicable, and therefore again it is predicable 
of itself. This is a contradiction, which shows that all the referents 
considered have no exclusive common predicate, and therefore, if defining 
predicates are essential to classes, do not form a class. 



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98 The Indefinables of Mathematics [chap, ix 

The matter may be put otherwise. In defining the would-be class of 
predicates, all those not predicable of themselves have been used up. 
The common predicate of all these predicates cannot be one of them, 
since for each of them there is at least one predicate (namely itself) of 
which it is not predicable. But again, the supposed common predicate 
cannot be any other predicate, for if it were, it would be predicable of 
itself, i.e. it would be a member of the supposed class of predicates, since 
these were defined as those of which it is predicable. Thus no predicate 
is left over which could attach to all the predicates considered. 

It follows from the above that not every definable collection of 
terms forms a class defined by a common predicate. This fact must be 
borne in mind, and we must endeavour to discover what properties a 
collection must have in order to form such a class. The exact point 
established by the above contradiction may be stated as follows : A pro- 
position apparently containing only one variable may not be equivalent 
to any proposition asserting that the variable in question has a certain 
predicate. It remains an open question whether every class must have 
a defining predicate. 

That all terms having a given relation to a given term form a class 
defined by an exclusive common predicate results from the doctrine of 
Chapter vii, that the proposition aRh can be analyzed into the subject 
a and the assertion Rh. To be a term of which Rh can be asserted 
appears to be plainly a predicate. But it does not follow, I think, 
that to be a term of which, for some value of «/, Ry can be asserted, is 
a predicate. The doctrine of propositional functions requires, however, 
that all terms having the latter property should form a class. This 
class I shall call the domain of the relation R as well as the class of 
referents. The domain of the converse relation will be also called the 
converse domain, as well as the class of relata. The two domains 
together will be called the field of the relation— a notion chiefly im- 
portant as regards series. Thus if paternity be the relation, fathers form 
its domain, children its converse domain, and fathers and children 
together its field. 

It may be doubted whether a proposition aRh can be regarded as 

asserting aR of &, or whether only Ra can be asserted oi h. In other 
words, is a relational proposition only an assertion concerning the 
referent, or also an assertion concerning the relatum.? If we take the 
latter view, we shall have, connected with (say) "ah greater than &,'' 
four assertions, namely " is greater than 5,'' " a is greater than," " is less 
than<^'' and "h is less than." I am inclined myself to adopt this view, 
but I know of no argument on either side. 

97. We can form the logical sum and product of two relations or 
of a class of relations exactly as in the case of classes, except that here 
we have to deal with double variability. In addition to these ways of 
combination, we have also the relative product, which is in general non- 



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96-99] Relations 99 

commutative, and therefore requires that the number of factors should 
be finite. If R^ S be two relations, to say that their relative product 
RS holds between two terms oc^ z is to say that there is a term ?/ to 
which oj has the relation i?, and which itself has the relation S to z. Thus 
brother-in-law is the relative product of wife and brother or of sister 
and husband : father-in-law is the relative product of wife and father, 
whereas the relative product of father and wife is mother or step-mother. 

98. There is a temptation to regard a relation as definable in 
extension as a class of couples. This has the formal advantage that it 
avoids the necessity for the primitive proposition asserting that every 
couple has a relation holding between no other pair of terms. But it is 
necessary to give sense to the couple, to distinguish the referent from the 
relatum : thus a couple becomes essentially distinct from a class of two 
terms, and must itself be introduced as a primitive idea. It would seem, 
viewing the matter philosophically, that sense can only be derived from 
some relational proposition, and that the assertion that a is referent and 
b relatum already involves a purely relational proposition in which a and 
b are terms, though the relation asserted is only the general one of 
referent to relatum. There are, in fact, concepts such as greater, which 
occur otherwise than as terms in propositions having two terms (§§ 48, 54); 
and no doctrine of couples can evade such propositions. It seems there- 
fore more correct to take an intensional view of relations, and to identify 
them rather with class -concepts than with classes. This procedure is 
formally more convenient, and seems also nearer to the logical facts. 
Throughout Mathematics there is the same rather curious relation of 
intensional and extensional points of view : the symbols other than 
variable terms {i.e. the variable class-concepts and relations) stand for 
intensions, while the actual objects dealt with are always extensions. 
Thus in the calculus of relations, it is classes of couples that are relevant, 
but the symbolism deals with them by means of relations. This is 
precisely similar to the state of things explained in relation to classes, 
and it seems unnecessary to repeat the explanations at length. 

99. Mr Bradley, in Appearance and Reality, Chapter iii, has based 
an argument against the reality of relations upon the endless regress 
arising from the fact that a relation which relates two terms must 
be related to each of 'them. The endless regress is undeniable, if 
relational propositions are taken to be ultimate, but it is very doubtful 
whether it forms any logical difficulty. We have already had occasion 
(§ 55) to distinguish two kinds of regress, the one proceeding merely to 
perpetually new implied propositions, the other in the meaning of a 
proposition itself; of these two kinds, we agreed that the former, since 
the solution of the problem of infinity, has ceased to be objectionable, 
while the latter remains inadmissible. We have to inquire which kind 
of regress occurs in the present instance. It may be urged that it is 
part of the very meaning of a relational proposition that the relation 

7—2 



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100 The Indefinables of Mathematics [chap, ix 

involved should have to the terms the relation expressed in saying that 
it relates them, and that this is what makes the distinction, which we 
formerly (§ 54) left unexplained, between a relating relation and a relation 
in itself. It may be urged, however, against this view, that the assertion 
of a relation between the relation and the terms, though implied, is no 
part of the original proposition, and that a relating relation is dis- 
tinguished from a relation in itself by the indefinable element of assertion 
which distinguishes a proposition from a concept. Against this it 
might be retorted that, in the concept " difference of a and h^'' difference 
relates a and h just as much as in the proposition " a and h differ"; but 
to this it may be rejoined that we found the difference of a and &, except 
in so far as some specific point of difference may be in question, to be 
indistinguishable from bare difference. Thus it seems impossible to 
prove that the endless regress involved is of the objectionable kind. 
We may distinguish, I think, between "a exceeds &'' and "a is greater 
than &," though it would be absurd to deny that people usually mean 
the same thing by these two propositions. On the principle, from which 
I can see no escape, that every genuine word must have some meaning, 
the is and than must form part of " a is greater than 5," which thus 
contains more than two terms and a relation. The is seems to state 
that a has to greater the relation of referent, while the than states 
similarly that h has to greater the relation of relatum. But " a exceeds 
&'' may be held to express solely the relation of a to &, without in- 
cluding any of the implications of further relations. Hence we shall 
have to conclude that a relational proposition aRh does not include 
in its meaning any relation of a or 6 to J?, and that the endless regress, 
though undeniable, is logically quite harmless. With these remarks, 
we may leave the further theory of relations to later Parts of the present 
work. 



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CHAPTER X. 
THE CONTRADICTION. 

100. Before taking leave of fundamental questions, it is necessary 
to examine more in detail the singular contradiction, already mentioned, 
with regard to predicates not predicable of themselves. Before attempt- 
ing to solve this puzzle, it will be well to make some deductions connected 
with it, and to state it in various different forms. I may mention that I 
was led to it in the endeavour to reconcile Cantor's proof that there can 
be no greatest cardinal number with the very plausible supposition that 
the class of all terms (which we have seen to be essential to all formal 
propositions) has necessarily the greatest possible number of members*. 

Let w be a class-concept which can be asserted of itself, i.e. such that 
" K? is a w.'''' Instances are class -concept^ and the negations of ordinary 
class-concepts, e.g. not-man. Then {a) if to be contained in another class ^, 
since to is a w., Z£^ is a x^ ; consequently there is a term of v which is 
a class-concept that can be asserted of itself. Hence by contraposition, 
(/8) if u be a class-concept none of whose members are class-concepts 
that can be asserted of themselves, no class-concept contained in u can 
be asserted of itself. Hence further, (7) if it be any class-concept what- 
ever, and It the class-concept of those members of u which are not 
predicable of themselves, this class-concept is contained in itself, and 
none of its members are predicable of themselves ; hence by (/S) it is not 
predicable of itself. Thus ii is not a u\ and is therefore not a u; for 
the terms of u that are not terms of u' are all predicable of themselves, 
which u is not. Thus (S) if it be any class -concept whatever, there is a 
class-concept contained in u which is not a member oi u^ and is also one 
of those class-concepts that are not predicable of themselves. So far, our 
deductions seem scarcely open to question. But if we now take the last 
of them, and admit the class of those class-concepts that cannot be 
asserted of themselves, we find that this class must contain a class-concept 
not a member of itself and yet not belonging to the class in question. 

We may observe also that, in virtue of what we have proved in (^8), the 
class of class-concepts which cannot be asserted of themselves, which we 
* See Part V, Chap, xliii, § 344 if. 



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102 The Indefinables of Mathematics [chap, x 

will call xv^ contains as members of itself all its sub-classes, although it is 
easy to prove that every class has more sub -classes than terms. Again, 
if y be any term of tv^ and tv be the whole of w except y^ then w\ being 
a sub-class of w, is not a to but is a no^ and therefore is y. Hence each 
class-concept which is a term of tv has all other terms of w as its 
extension. It follows that the concept Ucycle is a teaspoon, and teaspoon 
is a bicycle. This is plainly absurd, and any number of similar 
absurdities can be proved. 

101. Let us leave these paradoxical consequences, and attempt the 
exact statement of the contradiction itself. We have first the statement 
in terms of predicates, which has been given already. If zf be a predicate, 
X may or may not be predicable of itself. Let us assume that "not- 
predi cable of oneself" is a predicate. Then to suppose either that this 
predicate is, or that it is not, predicable of itself, is self-contradictory. 
The conclusion, in this case, seems obvious: " not-predicable of oneself 
is not a predicate. 

Let us now state the same contradiction in terms of class-concepts. 
A class-concept may or may not be a term of its own extension. "Class- 
concept which is not a term of its own extension'' appears to be a class- 
concept. But if it is a term of its own extension, it is a class-concept 
which is not a term of its own extension, and vice versa. Thus we must 
conclude, against appearances, that "class-concept which is not a term of 
its own extension" is not a class-concept. 

In terms of classes the contradiction appears even more extraordinary. 
A class as one may be a term of itself as many. Thus the class of all 
classes is a class; the class of all the terms that are not men is not a man, 
and so on. Do all the classes that have this property form a class ? If 
so, is it as one a member of itself as many or not ? If it is, then it is 
one of the classes which, as ones, are not members of themselves as many, 
and vice versa. Thus we must conclude again that the classes which as 
ones are not members of themselves as many do not form a class — or 
rather, that they do not form a class as one, for the argument cannot 
show that they do not form a class as many. 

102. A similar result, which, however, does not lead to a contradic- 
tion, may be proved concerning any relation. Let i? be a relation, and 
consider the class tiv of terms which do not have the relation R to them- 
selves. Then it is impossible that there should be any term a to which 
all of them and no other terms have the relation R. For, if there were 
such a term, the propositional function " oo does not have the relation R 
to X " would be equivalent to " oo has the relation R to a." Substituting 
a for X throughout, which is legitimate since the equivalence is formal, 
we find a contradiction. When in place of R we put e, the relation of 
a term to a class-concept which can be asserted of it, we get the above 
contradiction. The reason that a contradiction emerges here is that 
we have taken it as an axiom that any propositional function containing 



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100-103] The Contradiction 103 

only one variable is equivalent to asserting membership of a class defined 
by the propositional function. Either this axiom, or the principle that 
every class can be taken as one term, is plainly false, and there is no 
fundamental objection to dropping either. But having dropped the 
former, the question arises : Which propositional functions define classes 
which are single terms as well as many, and which do not? And with 
this question our real difficulties begin. 

Any method by which we attempt to establish a one-one or many- 
one correlation of all terms and all propositional functions must omit at 
least one propositional function. Such a method would exist if all 
propositional functions could be expressed in the form ...ez^, since this 
form correlates u with . . . e?/. But the impossibility of any such correla- 
tion is proved as follows. Let (^^ be a propositional function correlated 
with oc ; then, if the correlation covers all terms, the denial of 4)x {oc) will 
be a propositional function, since it is a proposition for all values of x. 
But it cannot be included in the correlation ; for if it were correlated 
with a, 4>a C-^) would be equivalent, for all values of x^ to the denial of 
0^ {x) ; but this equivalence is impossible for the value a, since it makes 
(^a{^) equivalent to its own denial. It follows that there are more 
propositional functions than terms — a result which seems plainly impos- 
sible, although the proof is as convincing as any in Mathematics. We 
shall shortly see how the impossibility is removed by the doctrine of 
logical types. 

103. The first method which suggests itself is to seek an ambiguity 
in the notion of e. But in Chapter vi we distinguished the various 
meanings as far as any distinction seemed possible, and we have just 
seen that with each meaning the same contradiction emerges. Let us, 
however, attempt to state the contradiction throughout in terms of 
propositional functions. Every propositional function which is not null, 
we supposed, defines a class, and every class can certainly be defined by 
a propositional function. Thus to say that a class as one is not a 
member of itself as many is to say that the class as one does not satisfy 
the function by which itself as many is defined. Since all propositional 
functions except such as are null define classes, all will be used up, in 
considering all classes having the above property, except such as do not 
have the above property. If any propositional function were satisfied 
by every class having the above property, it would therefore necessarily 
be one satisfied also by the class iv of all such classes considered as a 
single term. Hence the class to^ does not itself belong to the class w, 
and therefore there must be some propositional function satisfied by the 
terms of w but not by xv itself. Thus the contradiction re-emerges, and 
we must suppose, either that there is no such entity as te^, or that there 
is no propositional function satisfied by its terms and by no others. 

It might be thought that a solution could be found by denying the 
legitimacy of variable propositional functions. If we denote by k^^ for 



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104 The Indefinables of Mathematics [chap, x 

the moment, the class of values satisfying (/>, our propositional function 
is the denial of ct){k^)^ where <^ is the variable. The doctrine of 
Chapter vii, that (j> is not a separable entity, might make such a variable 
seem illegitimate ; but this objection can be overcome by substitut- 
ing for (j) the class of propositions 0«r, or the relation of (56^ to ^. 
Moreover it is impossible to exclude variable propositional functions 
altogether. Wherever a variable class or a variable relation occurs, 
we have admitted a variable propositional function, which is thus 
essential to assertions about every class or about every relation. The 
definition of the domain of a relation, for example, and all the general 
propositions which constitute the calculus of relations, would be swept 
away by the refusal to allow this type of variation. Thus we require 
some further characteristic by which to distinguish two kinds of varia- 
tion. This characteristic is to be found, I think, in the independent 
variability of the function and the argument. In general, c/)^ is itself 
a function of two variables, cj) and w ; of these, either may be given a 
constant value, and either may be varied without reference to the other. 
But in the type of propositional functions we are considering in this 
Chapter, the argument is itself a function of the propositional function : 
instead of (j^a^, we have cf) {f((f))], where y((/)) is defined as a function of 
cjy. Thus when is varied, the argument of which <p is asserted is 
varied too. Thus " ^r is an .r '' is equivalent to : " cj) can be asserted of 
the class of terms satisfying ^," this class of terms being ^. If here 
(j) is varied, the argument is varied at the same time in a manner 
dependent upon the variation of (p. For this reason, cj) [f{(b)]^ though 
it is a definite proposition when <x' is assigned, is not a propositional 
function, in the ordinary sense, when x is variable. Propositional 
functions of this doubtful type may be called quadratic forms ^ because 
the variable enters into them in a way somewhat analogous to that in 
which, in Algebra, a variable appears in an expression of the second 
degree. 

104. Perhaps the best way to state the suggested solution is to say 
that, if a collection of terms can only be defined by a variable pro- 
positional function, then, though a class as many .may be admitted, 
a class as one must be denied. When so stated, it appears that propo- 
sitional functions may be varied, provided the resulting collection is 
never itself made into the subject in the original propositional function. 
In such cases there is only a class as many, not a class as one. We took 
it as axiomatic that the class as one is to be found wherever there is 
a class as many ; but this axiom need not be universally admitted, 
and appears to have been the source of the contradiction. By denying 
it, therefore, the whole difficulty will be overcome. 

A class as one, we shall say, is an object of the same type as its 
terms ; i.e. any propositional function (x) which is significant when one 
of the terms is substituted for 00 is also significant when the class as one 



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103-105] The Contradiction 105 

is substituted. But the class as one does not always exist, and the class 
as many is of a different type from the terms of the class, even when the 
class has only one term, i.e. there are propositional functions ^ {u) in 
which u may be the class as many, which are meaningless if, for u^ we 
substitute one of the terms of the class. And so " x is one among «r's " 
is not a proposition at all if the relation involved is that of a term to its 
class as many ; and this is the only relation of whose presence a pro- 
positional function always assures us. In this view, a class as many may 
be a logical subject, but in propositions of a different kind from those in 
which its terms are subjects ; of any object other than a single term, the 
question whether it is one or many will have different answers according 
to the proposition in which it occurs. Thus we have " Socrates is one 
among men,^' in which men are plural ; but " men are one among species 
of animals," in which men are singular. It is the distinction of logical 
types that is the key to the whole mystery*. 

105. Other ways of evading the contradiction, which might be 
suggested, appear undesirable, on the ground that they destroy too 
many quite necessary kinds of propositions. It might be suggested 
that identity is introduced in " oc is not an <:r " in a way which is not 
permissible. But it has been already shown that relations of terms 
to themselves are unavoidable, and it may be observed that suicides 
or self-made men or the heroes of Smiles's Self-Help are all defined 
by relations to themselves. And generally, identity enters in a very 
similar way into formal implication, so that it is quite impossible to 
reject it. 

A natural suggestion for escaping from the contradiction would be 
to demur to the notion of a/Z terms or of all classes. It might be 
urged that no such sum-total is conceivable ; and if all indicates a whole, 
our escape from the contradiction requires us to admit this. But we 
have already abundantly seen that if this view were maintained against 
any term, all formal truth would be impossible, and Mathematics, whose 
characteristic is the statement of truths concerning any term, would be 
abolished at one stroke. Thus the correct statement of formal truths 
requires the notion of any term or every term, but not the collective 
notion of all terms. 

It should be observed, finally, that no peculiar philosophy is involved 
in the above contradiction, which springs directly from common sense, 
and can only be solved by abandoning some common-sense assumption. 
Only the Hegelian philosophy, which nourishes itself on contradictions, 
can remain indifferent, because it finds similar problems everywhere. In 
any other doctrine, so direct a challenge demands an answer, on pain 
of a confession of impotence. Fortunately, no other similar difficulty, 
so far as I know, occurs in any other portion of the Principles of 
Mathematics. 

^ On this subject,, see Appendix. 



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106 The Indefinables of Mathematics [chap, x 

106. We may now briefly review the conclusions arrived at in 
Part I. Pure Mathematics was defined as the class of propositions 
asserting formal implications and containing no constants except logical 
constants. And logical constants are : Implication, the relation of a 
term to a class of which it is a member, the notion of such ihat^ the 
notion of relation, and such further notions as are involved in formal 
implication, which we found (§ 93) to be the following : propositional 
function, class *, denoting, and any or every term. This definition brought 
Mathematics into very close relation to Logic, and made it practically 
identical with Symbolic Logic. An examination of Symbolic Logic justi- 
fied the above enumeration of mathematical indefinables. In Chapter iii 
we distinguished implication and formal implication. The former holds 
between any two propositions provided the first be false or the second 
true. The latter is not a relation, but the assertion, for every value 
of the variable or variables, of a propositional function which, for every 
value of the variable or variables, asserts an implication. Chapter iv 
distinguished what may be called things from predicates and relations 
(including the is of predications among relations for this purpose). It 
was shown that this distinction is connected with the doctrine of 
substance and attributes, but does not lead to the traditional results. 
Chapters v and vi developed the theory of predicates. In the former 
of these chapters it was shown that certain concepts, derived from 
predicates, occur in propositions not about themselves, but about com- 
binations of terms, such as are indicated by a//, every^ any^ a^ some^ and 
the. Concepts of this kind, we found, are fundamental in Mathematics, 
and enable us to deal with infinite classes by means of propositions of 
finite complexity. In Chapter vi we distinguished predicates, class- 
concepts, concepts of classes, classes as many, and classes as one. We 
agreed that single terms, or such combinations as result from and^ are 
classes, the latter being classes as many ; and that classes as many 
are the objects denoted by concepts of classes, which are the plurals 
of class-concepts. But in the present chapter we decided that it is 
necessary to distinguish a single term from the class whose only member 
it is, and that consequently the null-class may be admitted. 

In Chapter vii we resumed the study of the verb. Subject-predicate 
propositions, and such as express a fixed relation to a fixed term, could 
be analyzed, we found, into a subject and an assertion ; but this analysis 
becomes impossible when a given term enters into a proposition in a 
more complicated manner than as referent of a relation. Hence it 
became necessary to take propositional function as a primitive notion. 
A propositional function of one variable is any proposition of a set 
defined by the variation of a single term, while the other terms remain 

■^ The notion of class in general^ we decided, could be replaced^ as an indefinable_, 
by that of the class of propositions defined by a propositional function. 



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106] The Contradiction 107 

constant. But in general it is impossible to define or isolate the 
constant element in a propositional function, since what remains, when 
a certain term, wherever it occurs, is left out of a proposition, is in 
general no discoverable kind of entity. Thus the term in question 
must be not simply omitted, but replaced by a variable. 

The notion of the variable, we found, is exceedingly complicated. 
The o) is not simply any term, but any term with a certain individuality; 
for if not, any two variables would be indistinguishable. We agreed 
that a variable is any term qua term in a certain propositional function, 
and that variables are distinguished by the propositional functions in 
which they occur, or, in the case of several variables, by the place they 
occupy in a given multiply variable propositional function. A variable, 
we said, is the term in any proposition of the set denoted by a given 
propositional function. 

Chapter ix pointed out that relational propositions are ultimate, 
and that they all have sense : i.e, the relation being the concept as such 
in a proposition with two terms, there is another proposition containing 
the same terms and the same concept as such, as in ^' A is greater 
than B'''' and '' B is greater than A^ These two propositions, though 
different, contain precisely the same constituents. This is a characteristic 
of relations, and an instance of the loss resulting from analysis. Rela- 
tions, we agreed, are to be taken intensionally, not as classes of couples*. 

Finally, in the present chapter, we examined the contradiction re- 
sulting from the apparent fact that, if w be the class of all classes which 
as single terms are not members of themselves as many, then w as one 
can be proved both to be and not to be a member of itself as many. 
The solution suggested was that it is necessary to distinguish various 
types of objects, namely terms, classes of terms, classes of classes, classes 
of couples of terms, and so on; and that a propositional function^<r in 
general requires, if it is to have any meaning, that x should belong to 
some one type. Thus xeoo was held to be meaningless, because e requires 
that the relatum should be a class composed of objects which are of the 
type of the referent. The class as one, where it exists, is, we said, of the 
same type as its constituents ; but a quadratic propositional function in 
general appears to define only a class as many, and the contradiction 
proves that the class as one, if it ever exists, is certainly sometimes 
absent. 

* On this pointy however^ see Appendix. 



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PART II. 

NUMBER. 



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CHAPTEE XL 



DEFINITION OF CARDINAL NUMBERS. 

107. We have now briefly reviewed the apparatus of general logical 
notions with which Mathematics operates. In the present Part, it is to 
be shown how this apparatus suffices, without new indeflnables or new 
postulates, to establish the whole theory of cardinal integers as a special 
branch of Logic*. No mathematical subject has made, in recent years, 
greater advances than the theory of Arithmetic. The movement in 
favour of correctness in deduction, inaugurated by Weierstrass, has been 
brilliantly continued by Dedekind, Cantor, Frege, and Peano, and attains 
what seems its final goal by means of the logic of relations. As the 
modei-n mathematical theory is but imperfectly known even by most 
mathematicians, I shall begin this Part by four chapters setting forth 
its outlines in a non-symbolic form. I shall then examine the process 
of deduction from a philosophical standpoint, in order to discover, if 
possible, whether any unperceived assumptions have covertly intruded 
themselves in the course of the argument. 

108. It is often held that both number and particular numbers are 
indefinable. Now definability is a word which, in Mathematics, has a 
precise sense, though one which is relative to some given set of notions f. 
Given any set of notions, a term is definable by means of these notions 
when, and only when, it is the only term having to certain of these 
notions a certain relation which itself is one of the said notions. But 
philosophically, the word definition has not, as a rule, been employed in 
this sense ; it has, in fact, been restricted to the analysis of an idea 
into its constituents. This usage is inconvenient and, I think, useless ; 
moreover it seems to overlook the fact that wholes are not^ as a 

* Cantor has shown that it is necessary to separate the study of Cardinal and 
Ordinal numbers^ which are distinct entities, of which the former are simpler, but of 
which both are essential to ordinary Mathematics. On Ordinal numbers,, cf. Chaps. 
XXIX, XXXVIII, infra. 

t See Peano, F. 1901, p. 6 fF. and Padoa, ^^The'orie Algebrique des Nombres 
Entiers," Congres, Vol. in, p. 314 ff. 



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112 Number [chap, xi 

rule, determinate when their constituents are given, but are themselves 
new entities (which may be in some sense simple), defined, in the 
mathematical sense, by certain relations to their constituents. I shaill 
therefore, in future, ignore the philosophical sense, and speak only of 
mathematical definability. I shall, however, restrict this notion more 
than is done by Professor Peano and his disciples. They hold that the 
various branches of Mathematics have various indefinables, by means of 
which the remaining ideas of the said subjects are defined. I hold-— . 
and it is an important part of my purpose to prove — that all Pure 
Mathematics (including Geometry and even rational Dynamics) contains 
only one set of indefinables, namely the fundamental logical concepts 
discussed in Part I. When the various logical constants have been 
enumerated, it is somewhat arbitrary which of them we regard as 
indefinable, though there are apparently some which must be indefinable 
in any theory. But my contention is, that the indefinables of Pure 
Mathematics are all of this kind, and that the presence of any other 
indefinables indicates that our subject belongs to Applied Mathematics, 
Moreover, of the three kinds of definition admitted by Peano — the 
nominal definition, the definition by postulates, and the definition by 
abstraction* — I recognize only the nominal: the others, it would seem, 
are only necessitated by Peano*'s refusal to regard relations as part of the 
fundamental apparatus of logic, and by his somewhat undue haste in 
regarding as an individual what is really a class. These remarks will be 
best explained by considering their application to the definition of 
cardinal numbers. 

109. It has been common in the past, among those who regarded 
numbers as definable, to make an exception as regards the number 1, 
and to define the remainder by its means. Thus 2 was 1 -f 1, 3 was 
£ + 1, and so on. -This method was only applicable to finite numbers, 
and made a tiresome difference between 1 and other numbers ; moreover 
the meaning of + was commonly not explained. We are able now-a- 
days to improve greatly upon this method. In the first place, since 
Cantor has shown how to deal with the infinite, it has become both. 
desirable and possible to deal with the fundamental properties of numbers^ 
in a way which is equally applicable to finite and infinite numbers. In 
the second place, the logical calculus has enabled us to give an exact 
definition of arithmetical addition ; and in the third place, it has become 
as easy to define and 1 as to define any other number. In order to 
explain how this is done, I shall first set forth the definition of numbers 
by abstraction ; I shall then point out formal defects in this definition, 
and replace it by a nominal definition. 

Numbers are, it will be admitted, applicable essentially to classes. 
It is true that, where the number is finite, individuals may be enumerated 

* Cf. Burali-Forti^ '^'^Sur les differentes definitions du nombre reel/' Congres, iii, 
p. 294 ff. 



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108, 109] Definition of Cardinal Numbers 113 

to make up the given number, and may be counted one by one without 
any mention of a class-concept. But all finite collections of individuals 
form classes, so that what results is after all the number of a class. 
And where the number is infinite, the individuals cannot be enumerated, 
but must be defined by intension, i.e. by some common property in 
virtue of which they form a class. Thus when any class-concept is 
given, there is a certain number of individuals to which this class-concept 
is applicable, and the number may therefore be regarded as a property 
of the class. Jt_is this view of numbers which has rendered possible the 
whole theory of infinity, since it relieves us of the necessity of enume- 
rating the individuals whose number is to be considered. This view 
depends fundamentally upon the notion of all^ the numerical conjunction 
as we agreed to call it (§ 59). All men^ for example, denotes men con- 
joined in a certain way ; and it is as thus denoted that they have a 
number. Similarly all numbers or all points denotes numbers or points 
conjoined in a certain way, and as thus conjoined numbers or points have 
a number. Numbers, then, are to be regarded as properties of classes. 

The next question is : Under what circumstances do two classes have 
the same number? The answer is, that they have the same number 
when their terms can be correlated one to one, so that any one term of 
either corresponds to one and only one term of the other. This requires 
that there should be some one-one relation whose domain is the one 
class and whose converse domain is the other class. Thus, for example, 
if in a community all the men and all the women are married, and 
polygamy and polyandry are forbidden, the number of men must be the 
same as the number of woni,en. It might be thought that a one-one 
relation could not be defined except by reference to the number 1. But 
this is not the case. A relation is one-one when, if x and x have the 
relation in question to ?/, then x and x are identical ; while if oc has the 
relation in question to y and y\ then y and y' are identical. Thus it is 
possible, without the notion of unity, to define what is meant by a one- 
one relation. But in order to provide for the case of two classes which 
have no terms, it is necessary to modify slightly the above account of 
what is meant by saying that two classes have the same number. For if 
there are no terms, the terms cannot be correlated one to one. We 
must say: Two classes have the same number when, and only when, there 
is a one-one relation whose domain includes the one class, and which is 
such that the class of correlates of the terms of the one class is identical 
with the other class. From this it appears that two classes having no 
terms have always the same number of terms ; for if we take any one- 
one relation whatever, its domain includes the null-class, and the class 
of correlates of the null-class is again the null-class. When two classes 
have the same number, they are said to be similar. 

Some readers may suppose that a definition of what is meant by 
saying that two classes have the same number is wholly unnecessary. 



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114 Number [chap, xi 

The way to find out, they may say, is to count both classes. It is such 
notions as this which have, until very recently, prevented the exhibition 
of Arithmetic as a branch of Pure Logic. For the question immediately 
arises : What is meant by counting ? To this question we usually get 
only some irrelevant psychological answer, as, that counting consists in 
successive acts of attention. In order to count 10, I suppose that ten 
acts of attention are required : certainly a most useful definition of the 
number 10 ! Counting has, in fact, a good meaning, which is not 
psychological. But this meaning is highly complex ; it is only applicable 
to classes which can be well-ordered, which are not known to be all 
classes ; and it only gives the number of the class when this number is 
Unite — a rare and exceptional case. We must not, therefore, bring in 
counting where the definition of numbers is in question. 

The relation of similarity between classes has the three properties of 
being reflexive, symmetrical, and transitive ; that is to say, if Uy v^ iv be 
classes, u is similar to itself; if u be similar to v^ v is similar to u\ and 
if u be similar to ^, and v to w^ then u is similar to i<d. These properties 
all follow easily from the definition. Now these three properties of a 
relation are held by Peano and common sense to indicate that when the 
relation holds between two terms, .those two terms have a certain common 
property, and vice versa. This common property we call their number*. 
This is the definition of numbers by abstraction. 

110. Now this definition by abstraction, and generally the process 
employed in such definitions, suffers from an absolutely fatal formal 
defect : it does not show that only one object satisfies the definition f. 
Thus instead of obtaining one common property of similar classes, which 
is the number of the classes in question, we obtain a class of such 
properties, with no means of deciding how many terms this class contains. 
In order to make this point clear, let us examine what is meant, in the 
present instance, by a common property. What is meant is, that any 
class has to a certain entity, its number, a relation which it has to nothing 
else, but which all similar classes (and no other entities) have to the said 
number. That is, there is a many-one relation which every class has to 
its number and to nothing else. Thus, so far as the definition by 
abstraction can show, any set of entities to each of which some class has 
a certain many-one relation, and to one and only one of which any given 
class has this relation, and which are such that all classes similar to a 
given class have this relation to one and the same entity of the set, 
appear as the set of numbers, and any entity of this set is the number of 
some class. If, then, there are many such sets of entities — and it is easy 

^- Cf. Peano, F. 1901, § 32, -0, Note. 

+ On the necessity of this condition^, cf. Padoa^, loc. cit., p. 824. Padoa appears 
not to perceive^ however^ that all definitions define the single individual of a class : 
when what is defined is a class, this must he the only term of some class of classes. 



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109-111] Definition of Cardinal Numbers 115 

to prove that there are an infinite number of them — every class will 
have many numbers, and the definition wholly fails to define iA^ number 
of a class. This argument is perfectly general, and shows that definition 
by abstraction is never a logically valid process. 

111. There are two ways in which we may attempt to remedy this 
defect. One of these consists in defining as the number of a class the 
whole class of entities, chosen one from each of the above sets of entities, 
to which all classes similar to the given class (and no others) have some 
many-one relation or other. But this method is practically useless, since 
all entities, without exception, belong to every such class, so that every 
class will have as its number the class of all entities of every sort and 
description. The other remedy is more practicable, and applies to all 
the cases in which Peano employs definition by abstraction. This 
method is, to define as the number of a class the class of all classes 
similar to the given class. Membership of this class of classes (considered 
as a predicate) is a common property of all the similar classes and of no 
others ; moreover every class of the set of similar classes has to the set 
a relation which it has to nothing else, and which every class has to its 
own set. Thus the conditions are completely fulfilled by this class of 
classes, and it has the merit of being determinate when a class is given, 
and of being different for two classes which are not similar. This, then, 
is an irreproachable definition of the number of a class in purely logical 
terms. 

To regard a number as a class of classes must appear, at first sight, 
a wholly indefensible paradox. Thus Peano {F. 1901, § 3S) remarks that 
" we cannot identify the number of [a class] a with the class of classes in 
question \i.e. the class of classes similar to a\ for these objects have 
different properties,*" He does not tell us what these properties are, and 
for my part I am unable to discover them. Probably it appeared to him 
immediately evident that a number is not a class of classes. But some- 
thing may be said to mitigate the appearance of paradox in this view. 
In the first place, such a word as couple or trio obviously does denote a 
class of classes. Thus what we have to say is, for example, that " two 
men"' means "logical product of class of men and couple," and "there are 
two men"" means "there is a class of men which is also a couple."' In the 
second place, when we remember that a class-concept is not itself a col- 
lection, but a property by which a collection is defined, we see that, if we 
define the number as the class-concept, not the class, a number is really 
defined as a common property of a set of similar classes and of nothing 
else., This view removes the appearance of paradox to a great degree. 
There is, however, a philosophical difficulty in this view, and generally in 
the connection of classes and predicates. It may be that there are many 
predicates common to a certain collection of objects and to no others. In 
this case, these predicates are all regarded by Symbolic Logic as equivalent, 
and any one of them is said to be equal to any other. Thus if the 

8—2 



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116 Number [chap, xi 

predicate were defined by the collection of objects, we should not obtain, 
in general, a single predicate, but a class of predicates; for this class of 
predicates we should require a new class-concept, and so on. The only 
available class-concept would be " predicability of the given collection of 
terms and of no others." But in the present case, where the collection is 
defined by a certain relation to one of its terms, there is some danger of 
a logical error. Let u be a class; then the number of u^ we said, is the 
class of classes" similar to u. But "similar to 2^" cannot be the actual 
concept which constitutes the number of u\ for, if v be similar to u^ 
"similar to v'''' defines the same class, although it is a different concept. 
Thus we require, as the defining predicate of the class of similar classes, 
some concept which does not have any special relation to one or more of 
the constituent classes. In regard to every particular number that may 
be mentioned, whether finite or infinite, such a predicate is, as a matter 
of fact, discoverable; but when all we are told about a number is that it 
is the number of some class u^ it is natural that a special reference to u 
should appear in the definition. This, however, is not the point at issue. 
The real point is, that what is defined is the same whether we use the 
predicate "similar to ^7 or "similar to v^"" provided u is similar to v. 
This shows that it is not the class-concept or defining predicate that is 
defined, but the class itself whose terms are the various classes which are 
similar to u or to t;. It is such classes, therefore, and not predicates such 
as "similar to z^,"' that must be taken to constitute numbers. 

Thus, to sum up: Mathematically, a number is nothing but a class of 
similar classes: this definition allows the deduction of all the usual 
properties of numbers, whether finite or infinite, and is the only one (so 
far as I know) which is possible in terms of the fundamental concepts of 
general logic. But philosophically we may^ admit that every collection 
of similar classes has some common predicate applicable to no entities 
except the classes in question, and if we can find, by inspection, that 
there is a certain class of such common predicates, of which one and only 
one applies to each collection of similar classes, then we may, if we see 
fit, call this particular class of predicates the class of numbers.^ For my 
part, I do not know whether there is any such class of predicates, and 
I do know that, if there be such a class, it is wholly irrelevant to Ma- 
thematics. Wherever Mathematics derives a common property from a 
reflexive, symmetrical, and transitive relation, all mathematical purposes 
of the supposed common property are completely served when it is 
replaced by the class of terms having the given relation to a given term ; 
and this is precisely the case presented by cardinal numbers. For the 
future, therefore, I shall adhere to the above definition, since it is at 
once precise and adequate to all mathematical uses. 



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CHAPTER XIL 



ADDITION AND MULTIPLICATION. 

112. Ix most mathematical accounts of arithmetical operations we 
find the error of endeavouring to give at once a definition which shall be 
applicable to rationals^ or even to real numbers, without dwelling at 
sufficient length upon the theory of integers. For the present, integers 
alone will occupy us. The definition of integers, given in the preceding 
chapter, obviously does not admit of extension to fractions ; and in fact 
the absolute difference between integers and fractions, even between 
integers and fractions whose denominator is unity, cannot possibly be too 
strongly emphasized. What rational fractions are, and what real numbers 
are, I shall endeavour to explain at a later stage; positive and negative 
numbers also are at present excluded. The integer^s with which we are 
now concerned are not positive, but signless. And so the addition and 
multiplication to be defined in this chapter are only applicable to integers ; 
but they have the merit of being equally applicable to finite and infinite 
integers. Indeed, for the present, I shall rigidly exclude all propositions 
which involve either the finitude or the infinity of the numbers considered, 

113. There is only one fundamental kind of addition, namely the 
logical kind. All other kinds can be defined in terms of this and logical 
multiplication. In the present chapter the addition of integers is to be 
defined by its means. Logical addition, as was explained in Part I, 
is the same as disjunction; if p and q are propositions, their logical 
sum is the proposition ^"^ p or <^,'" and if u and v are classes, their 
logical sum is the class "?^ or f," i.e. the class to w^iich belongs every 
term which either belongs to u or belongs to v. The logical sum 
of two classes u and v may be defined in terms of the logical product 
of two propositions, as the class of terms belonging to every 'class 
in which both u and v are contained*. This definition is not essen- 
tially confined to two classes, but may be extended to a class of 
classes, w^hether finite or infinite. Thus if A; be a class of classes, the 
logical sum of the classes composing I: (called for short the sum of A:) is 



F. 1901, § 2, Prop. 1 -0. 



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118 Number [chap, xn 

the class of terms belonging to every class which contains every class 
which is a term of h. It is this notion which underlies arithmetical 
addition. If A: be a class of classes no two of which have any common 
terms (called for short an exclusive class of classes), then the arith- 
metical sum of the numbers of the various classes of k is the number of 
terms in the logical sum of A:. This definition is absolutely general, and 
applies equally whether k or any of its constituent classes be finite 
or infinite. In order to assure ourselves that the resulting number 
depends only upon the numbers of the various classes belonging to k^ and 
not upon the particular class k that happens to be chosen, it is necessary 
to prove (as is easily done) that if k' be another exclusive class of classes, 
similar to k^ and every member of k is similar to its correlate in k\ and 
vice versa, then the number of terms in the sum of k is the same as the 
number in the sum of k\ Thus, for example, suppose k has only two 
terms, u and v, and suppose u and v have no common part. Then the 
number of terms in the logical sum of u and v is the sum of the number 
of terms in u and in v; and if ii be similar to w, and v' to v, and u\ v' 
have no common part, then the sum of u and v is similar to the 
sum of u and v. 

114. With regard to this definition of a sum of numbers, it is to be 
observed that it cannot be freed from reference to classes which have the 
numbers in question. The number obtained by summation is essentially 
the number of the logical sum of a certain class of classes or of some 
similar class of similar classes. The necessity of this reference to classes 
emerges when one number occurs twice or oftener in the summation. It 
is to be observed that the numbers concerned have no order of summation, 
so that we have no such proposition as the commutative law : this pro- 
position, as introduced in Arithmetic, results only from a defective 
symbolism, which causes an order among the symbols which has no 
correlative order in what is symbolized. But owing to the absence of 
order, if one number occurs twice in a summation, we cannot distinguish 
a first and a second occurrence of .the said number. If we exclude a 
reference to classes which have the said number, there is no sense in the 
supposition of its occurring twice : the summation of a class of numbers 
can be defined, but in that case, no number can be repeated. In the 
above definition of a sum, the numbers concerned are defined as the 
numbers of certain classes, and therefore it is not necessary to decide 
whether any number is repeated or not. But in order to define, without 
reference to particular classes, a sum of numbers of which some are 
repeated, it is necessary first to define multiplication. 

This point may be made clearer by considering a special case, such as 
1 -f 1. It is plain that we cannot take the number 1 itself twice over, 
for there is one number 1, and there are not two instances of it. And if 
the logical addition of 1 to itself were in question, we should find that 
1 and 1 is 1, according to the general principle of Symbolic Lof 'c. Nor 



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113-116] Addition and Multiplication 119 

can we define 1 + 1 as the arithmetical sum of a certain class of numbers. 
This method can be employed as regards 1 + ^, or any sum in which no 
number is repeated; but as regards 1 + 1, the only class of numbers 
involved is the class whose only member is 1, and since this class has one 
member, not two, we cannot define 1 + 1 by its means. Thus the full 
definition of 1 + 1 is as follows : 1+1 is the number of a class xc which 
is the logical sum of two classes u and v which have no common term 
and have each only one term. The chief point to be observed is, that 
logical addition of classes is the fundamental notion, while the arith- 
metical addition of numbers is wholly subsequent. 

115. The general definition of multiplication is due to Mr A. N. 
Whitehead*. It is as follows. Let A; be a class of classes, no two of 
which have any term in common. Form what is called the multiplicative 
class of h^ I.e. the class each of whose terms is a class formed by choosing 
one and only one term from each of the classes belonging to k. Then 
the number of terms in the multiplicative class of k is the product of all 
the numbers of the various classes composing k. This definition, like 
that of addition given above, has two merits, which make it preferable 
to any other hitherto suggested. In the first place, it introduces no 
order among the numbers nmltiplied, so that there is no need of the 
commutative law, which, here as in the case of addition, is concerned 
rather with the symbols than with what is symbolized. In the second 
place, the above definition does not require us to decide, concerning any 
of the numbers involved, whether they are finite or infinite. Cantor has 
given t definitions of the sum and product of two numbers, which do not 
require a decision as to whether these numbers are finite or infinite. 
These definitions can be extended to the simi and product of any Jinite 
number of finite or infinite numbers ; but they do not, as they stand, 
allow the definition of the sum or product of an infinite number of 
numbers. This grave defect is remedied in the above definitions, which 
enable us to pursue Arithmetic, as it ought to be pursued, without 
introducing the distinction of finite and infinite until we wish to study 
it. Cantor's definitions have also the formal defect of introducing an 
order among the numbers summed or multiplied : but this is, in his 
case, a mere defect in the symbols chosen, not in the ideas which he 
symbolizes. Moreover it is not practically desirable, in the case of the 
sum or product of two numbers, to avoid this formal defect, since the 
resulting cumbrousness becomes intolerable. 

116. It is easy to deduce from the above definitions the usual 
connection of addition and multiplication, which may be thus stated. 
If k be a class of b mutually exclusive classes, each of which contains 
a terms, then the logical sum of k contains a x b terms ;[:. It is also 

* American Journal of Mathematics, Oct. 1902. 

t Math. Annalen, Vol. xlvi^ § 3. % See \^Tiitehead^ loc. cit. 



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120 Number [chap, xii 

easy to obtain the definition of a*, and to prove the associative and dis- 
tributive laws, and the formal laws for powers, such as cd^a^ = c^^^. But 
it is to be observed that exponentiation is not to be regarded as a new 
independent operation, since it is merely an application of multipli- 
cation. It is true that exponentiation can be independently defined, 
as is done by Cantor*, but there is no advantage in so doing. Moreover 
exponentiation unavoidably introduces ordinal notions, since a^ is not in 
general equal to If. For this reason we cannot define the result of an 
infinite number of exponentiations. Powers, therefore, are to be regarded 
simply as abbreviations for products in which all the numbers multiplied 
together are equal. 

From the data which we now possess, all those propositions which 
hold equally of finite and infinite numbers can be deduced. The next 
step, therefore, is to consider the distinction between the finite and the 
infinite. 

■^ Loc. cit,, § 4. 



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CHAPTER XIII. 



FINITE AND INFINITE. 

117. The purpose of the present chapter is not to discuss the philo- 
sophical difficulties concerning the infinite, which are postponed to 
Part V. For the present I wish merely to set forth briefly the mathe- 
matical theory of finite and infinite as it appears in the theory of 
cardinal numbers. This is its most fundamental form, and must be 
understood before the ordinal infinite can be adequately explained*. 

Let u be any class, and let u be a class formed by taking away one 
term x from u. Then it may or may not happen that u is similar to u . 
For example, if it be the class of all finite numbers, and v! the class of 
all finite numbers except 0, the terms oiii are obtained by adding 1 to each 
of the terms of u^ and this correlates one term of n with one of it and vice 
versd^ no term of either being omitted or taken twice over. Thus ri is 
similar to u. But if u consists of all finite numbers up to ?2, where n is 
some finite number, and it consists of all these except 0, then u is not 
similar to u. If there is one term cc which can be taken away from it to 
leave a similar class u\ it is easily proved that if any other term y is 
taken away instead of oc we also get a class similar to u. When it is 
possible to take away one term from u and leave a class u similar to w, 
we say that u is an infinite class. When this is not possible, we say that 
u is di finite class. From these definitions it follows that the null-class is 
finite, since no term can be taken from it. It is also easy to prove that 
if u be a finite class, the class formed by adding one term to it is finite ; 
and conversely if this class is finite, so is u. It follows from the definition 
that the numbers of finite classes other than the null-class are altered 
by subtracting 1, while those of infinite classes are unaltered by this 
operation. It is easy to prove that the same holds of the addition of 1. 

118. Among finite classes, if one is a proper part of another, the 
one has a smaller number of terms than the other. (A proper part is 
a part not the whole.) But among infinite classes, this no longer holds. 

* On the present topic cf. Cantor^ Math. Annalen, Vol. xlvi, §§ 5^ 6, where 
most of M'hat follows will be found. 



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122 Number [chap, xm 

This distinction is, in fact, an essential part of the above definitions of 
the finite and the infinite. Of two infinite classes, one may have a 
greater or a smaller number of terms than the other. A class u is said 
to be greater than a class t?, or to have a number greater than that of t;, 
when the two are not similar, but v is similar to a proper part of it. It 
is known that if u is similar to a proper part of v^ and z; to a proper 
part of u (a case which can only arise when u and v are infinite), then u 
is similar to v\ hence "i^ is gi*eater than t;'" is inconsistent with " t^ is 
greater than ^^.'' It is not at present known whether, of two different 
infinite numbers, one must be greater and the other less. But it is known 
that there is a least infinite number, i.e. a number which is less than any 
different infinite number. This is the number of finite integers, which 
will be denoted, in the present work, by tto*- This number is capable of 
several definitions in which no mention is made of the finite numbers. In 
the first place it may be defined (as is implicitly done by Can tor f) by means 
of the principle of mathematical induction. This definition is as follows: 
tto is the number of any class u which is the domain of a one-one relation 
jK, whose converse domain is contained in but not coextensive with u^ 
and which is such that, calling the term to which x has the relation R 
the siwcessor of x^ if s be any class to which belongs a term of u which is 
not a successor of any other term of ii^ and to which belongs the successor 
of every term of u which belongs to ,9, then every term of u belongs to s. 
Or again, we may define ao as follows. Let P be a transitive and asym- 
metrical relation, and let any two different terms of the field of P have the 
relation P or its converse. Further let any class u contained in the field 
of P and having successors {i.e. terms to which every term of u has the 
relation P) have an immediate successor, i.e. a term whose predecessors 
either belong to u or precede some term of u ; let there be one term of 
the field of P which has no predecessors, but let every term which has 
predecessors have successors and also have an immediate predecessor ; 
then the number of terms in the field of P is a^. Other definitions may 
be suggested, but as all are equivalent it is not necessary to multiply 
them. The following characteristic is important : Every class whose 
number is ao can be arranged in a series having consecutive terms, a 
beginning but no end, and such that the number of predecessors of any 
term of the series is finite ; and any series having these characteristics 
has the number ao- 

It is very easy to show that every infinite class contains classes whose 
number is cl^. For let u be such a class, and let x^ be a term of u. 
Then u is similar to the class obtained by taking away .Tq, which ve will 
call the class u^. Thus 11^ is an infinite class. From this we can take 



* Cantor employs for this number the Hebrew Aleph with the suffix 0, but this 
notation is inconvenient. 

t Math. Annalen, Vol. xlvi, § 6. 



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118, 119] Finite and Infinite 123 

away a term ^i, leaving an infinite class ?/./, and so on. The series of 
terms ^'i, x^^... is contained in ii^ and is of the type which has the 
number ^o. From this point we can advance to an alternative definition 
of the finite and the infinite by means of mathematical induction, which 
must now be explained. 

119. If n be any finite number, the number obtained by adding 
1 to n is also finite, and is different from n. Thus beginning with 
we can form a series of numbers by successive additions of 1. We 
may define finite numbers, if we choose, as those numbers that can be 
obtained from by such steps, and that obey mathematical induction. 
That is, the class of finite numbers is the class of numbers which is 
contained in every class s to which belongs and the successor of every 
number belonging to 5, where the successor of a number is the number 
obtained by adding 1 to the given number. Now a^ is not such a 
number, since, in virtue of propositions already proved, no such number 
is similar to a part of itself. Hence also no number greater than a^ 
is finite according to the new definition. But it is easy to prove that 
every number less than a^ is finite with the new definition as with the 
old. Hence the two definitions are equivalent. Thus we may define 
finite numbers either as those that can be reached by mathematical 
induction, starting from and increasing by 1 at each step, or as those 
of classes which are not similar to the parts of themselves obtained by 
taking away single terms. These two definitions are both frequently 
employed, and it is important to realize that either is a consequence 
of the other. Both will occupy us much hereafter; for the present 
it is only intended, without controversy, to set forth the bare outlines 
of the mathematical theory of finite and infinite, leaving the details to 
be filled in during the course of the work. 



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CHAPTER XIV. 



THEORY OF FINITE NUMBERS. 

120. Having now clearly distinguished the finite from the infinite, 
we can devote ourselves to the consideration of finite numbers. Tt is 
customary, in the best treatises on the elements of Arithmetic*, not to 
define number or particular finite numbers, but to begin with certain 
axioms or primitive propositions, from which all the ordinary results 
are shown to follow. This method makes Arithmetic into an in- 
dependent study, instead of regarding it, as is done in the present 
work, as merely a development, without new axioms or indefinables, of a 
certain branch of general Logic. For this reason, the method in question 
seems to indicate a less degree of analysis than that adopted here. I 
shall nevertheless begin by an exposition of the more usual method, 
and then proceed to definitions and proofs of what are usually taken 
as indefinables and indemonstrables. For this purpose, I shall take 
Peano's exposition in the Forinulaire^^ which is, so far as I know, 
the best from the point of view of accuracy and rigour. This exposition 
has the inestimable merit of showing that all Arithmetic can be de- 
veloped from three fundamental notions (in addition to those of general 
Logic) and five fundamental propositions concerning these notions. It 
proves also that, if the three notions be regarded as determined by the 
five propositions, these five propositions are mutally independent. This 
is shown by finding, for each set of four out of the five propositions, 
an intei-pretation which renders the remaining proposition false. It 
therefore only remains, in order to connect Peano's theory with that 
here adopted, to give a definition of the three fundamental notions and 
a demonstration of the five fundamental propositions. When once this 
has been accomplished, we know with certainty that everything in the 
theory of finite integers follows. 

* Except Frege's Grandgesetze der AHthmetik (Jena, 1893). 

t F. 1901, Part II and F. 1899, § 20 ff. F. 1901 diifers from earlier editions in 
making /* number is a class " a primitive proposition. I regard this as unnecessary, 
since it is implied by ^'0 is a number." I therefore follow the earlier editions. 



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120-122] Theory of Finite Numbers 125 

Peano's three indefinables are 0, finite integer*'^ and successor of. 
It is assumed, as part of the idea of succession (though it would, 
I think, be better to state it as a separate axiom), that every number 
has one and only one successor. (By successor is meant, of course, 
immediate successor.) Peano's primitive propositions are then the 
following. (1) is a number. (S) If a is a number, the successor of 
a is a number. (8) If two numbers have the same successor, the two 
numbers are identical. (4) is not the successor of any number. 
(5) If s be a class to which belongs and also the successor of every 
number belonging to *, then every number belongs to s. The last of 
these propositions is the principle of mathematical induction. 

121. The mutual independence of these five propositions has been 
demonstrated by Peano and Padoa as follows f. (1) Giving the usual 
meanings to and successor^ but denoting by number finite integers 
other than 0, all the above propositions except the first are true. 

(2) Giving the usual meanings to and successor^ but denoting by 
number only finite integers less than 10, or less than any other specified 
finite integer, all the above propositions are true except the second. 

(3) A series which begins by an antiperiod and then becomes periodic 
(for example, the digits in a decimal which becomes recurring after a 
certain number of places) will satisfy all the above propositions except 
the third. (4) A periodic series (such as the hours on the clock) 
satisfies all except the fourth of the primitive propositions, (5) Giving 
to successor the meaning greater by 2, so that the successor of is 2, 
and of 2 is 4, and so on, all the primitive propositions are satisfied 
except the fifth, which is not satisfied if s be the class of even numbers 
including 0. Thus no one of the five primitive propositions can be 
deduced from the other four. 

122. Peano points out Qoc. cit.) that other classes besides that of 
the finite integers satisfy the above five propositions. What he says 
is as follows : " There is an infinity of systems satisfying all the primitive 
propositions. They are all verified, e.g., by replacing number and by 
number other than and 1. All the systems which satisfy the primitive 
propositions have a one -one correspondence with the numbers. Number 
is what is obtained from all these systems by abstraction ; in other 
words, number is the system which has all the properties enunciated 
in the primitive propositions, and those only." This observation appears 
to me lacking in logical correctness. In the first place, the question 
arises : How are the various systems distinguished, which agree in satis- 
fying the primitive propositions ? How, for example, is the system 
beginning with 1 distinguished from that beginning with 0? To this 

■^ Throughout the rest of this chapter^ I shall use number as synonymous with 
finite integer. 

t F. 1899, p. 30. 



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126 Number [char xiv 

question two different answers may be given. We may say that and 
1 are both primitive ideas, or at least that is so, and that therefore 
and 1 can be intrinsically distinguished, as yellow and blue are dis- 
tinguished. But if we take this view — which, by the way, will have to 
be extended to the other primitive ideas, number and succession — we 
shall have to say that these three notions are what I call constants, 
and that there is no need of any such process of abstraction as Peano 
speaks of in the definition of number. In this method, 0, number, and 
succession appear, like other indefinables, as ideas which must be simply 
recognized. Their recognition yields what mathematicians call the 
existence- theorem, I.e. it assures us that there really are numbers. 
But this process leaves it doubtful whether numbers are logical constants 
or not, and therefore makes Arithmetic, according to the definition in 
Part I, Chapter i^ prima fade a branch of Applied Mathematics. More- 
over it is evidently not the process which Peano has in mind. The 
other answer to the question consists in regarding 0, number, and 
succession as a class of three ideas belonging to a certain class of trios 
defined by the five primitive propositions. It is very easy so to state 
the matter that the five primitive propositions become transformed into 
the nominal definition of a certain class of trios. There are then no 
longer any indefinables or indemonstrables in our theory, which has 
become a pure piece of Logic. But 0, number and succession become 
variables, since they are only determined as one of the class of trios : 
moreover the existence-theorem now becomes doubtful, since we cannot 
know, except by the discovery of at least one actual trio of this class, 
that there are any such trios at all. One actual trio, however, would 
be a constant, and thus we require some method of giving constant 
values to 0, number, and succession. What we can show is that, if there 
is one such trio, there are an infinite number of them. For by striking 
out the first term from any class satisfying the conditions laid down 
concerning number, we always obtain a class which again satisfies the 
conditions in question. But even this statement, since the meaning of 
number is still in question, must be differently worded if circularity 
is to be avoided. Moreover we must ask ourselves : Is any process of 
abstraction from all systems satisfying the five axioms, such as Peano 
contemplates, logically possible "^ Every term of a class is the term it 
is, and satisfies some proposition which becomes false when another term 
of the class is substituted. There is therefore no term of a class which 
has merely the properties defining the class and no others. What 
Peano*'s process of abstraction really amounts to is the consideration of 
the class and variable members of it, to the exclusion of constant 
members. For only a variable member of the class will have only the 
properties by which the class is defined. Thus Peano does not succeed 
in indicating any constant meaning for 0, number, and succession, nor 
in showing that any constant meaning is possible, since the ex'stence- 



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122, 123] Theory of Finite Numbers 127 

theorem is not proved. His only method, therefore, is to say that at 
least one such constant meaning can be immediately perceived, but is 
not definable. This method is not logically unsound, but it is wholly 
different from the impossible abstraction which he suggests. And the 
proof of the mutual independence of his five primitive propositions is 
only necessary in order to show that the definition of the class of trios 
determined by them is not redundant. Redundancy is not a logical 
error, but merely a defect of what may be called style. My object, in 
the above account of cardinal numbers, has been to prove, from general 
Logic, that there is one constant meaning which satisfies the above five 
propositions, and that this constant meaning should be called number, 
or rather finite cardinal number. And in this way, new indefinables 
and indemonstrables are wholly avoided ; for when we have shown that 
the class of trios in question has at least one member, and when this 
member has been used to define number, we easily show that the class 
of trios has an infinite number of members, and we define the class 
by means of the five properties enumerated in Peano's primitive proposi- 
tions. For the comprehension of the connection between Mathematics 
and Logic, this point is of very great importance, and similar points will 
occur constantly throughout the present work. 

123. In order to bring out more clearly the difference between 
Peano''s procedure and mine, I shall here repeat the definition of the 
class satisfying his five primitive propositions, the definition of finite 
number^ and the proof, in the case of finite numbers, of his five primitive 
propositions. 

The class of classes satisfying his axioms is the same as the class of 
classes whose cardinal number is a,,? ^•^' the class of classes, according to 
my theory, which is a^. It is most simply defined as follows : a^ is the 
class of classes u each of which is the domain of some one-one relation R 
(the relation of a term to its successor) which is such that there is at 
least one term which succeeds no other term, every term which succeeds 
has a successor, and ii is contained in any class s which contains a term 
of u having no predecessors, and also contains the successor of every 
term of u which belongs to s. This definition includes Peano's five 
primitive propositions and no more. Thus of every such class all the 
usual propositions in the arithmetic of finite numbers can be proved : 
addition, multiplication, fractions, etc. can be defined, and the whole of 
analysis can be developed, in so far as complex numbers are not involved. 
But in this whole development, the meaning of the entities and relations 
which occur is to a certain degree indeterminate, since the entities and 
the relation with which we start are variable members of a certain class. 
Moreover, in this whole development, nothing shows that there are such 
classes as the definition speaks of. 

In the logical theory of cardinals, we start from the opposite end. 
We first define a certain class of entities, and then show that this class 



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128 Number [chap, xiv 

of entities belongs to the class Oq above defined. This is done as follows. 
(1) is the class of classes whose only member is the null-class. (2) A 
number is the class of all classes similar to any one of themselves. (S) 1 is 
the class of all classes which are not null and are such that, if x belongs to 
the class, the class without oc is the null-class ; or such that, if x and y 
belong to the class, then x and y are identical. (4) Having shown that 
if two classes be similar, and a class of one term be added to each, the 
sums are similar, we define that, if n be a number, 7?- + 1 is the number 
resulting from adding a unit to a class of n terms. (5) Finite numbers 
are those belonging to every class s to which belongs 0, and to which 
n-{-l belongs if n belongs. This completes the definition of finite 
numbers. We then have, as regards the five propositions which Peano 
assumes : (1) is a number. (S) Meaning n-\-\ by the successor of n^ 
if n be a number, then n -h 1 is a number. (3) If tz + 1 = 7/^ -f 1? then 
n = m. (4) If n be any number, 7^ -f 1 is different from 0. (5) If ^ be 
a class, and belongs to this class, and if when n belongs to it, n -f 1 
belongs to it, then all finite numbers belong to it. Thus all the five 
essential properties are satisfied by the class of finite numbers as above 
defined. Hence the class of classes a^ has members, and the class finite 
number is one definite member of a^. There is, therefore, from the 
mathematical standpoint, no need whatever of new indefinables or 
indemonstrables in the whole of Arithmetic and Analysis. 



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CHAPTER XV. 



ADDITION OF TERMS AND ADDITION OF CLASSES. 

124. Having now briefly set forth the mathematical theory of 
cardinal numbers, it is time to turn our attention to the philosophical 
questions raised by this theory, I shall begin by a few preliminary 
remarks as to the distinction between philosophy and mathematics, and 
as to the function of philosophy in such a subject as the foundations of 
mathematics. The following observations are not necessarily to be 
regarded as applicable to other branches of philosophy, since they are 
derived specially from the consideration of the problems of logic. 

The distinction of philosophy and mathematics is broadly one of 
point of view : mathematics is constructive and deductive, philosophy i& 
critical, and in a certain impersonal sense controversial. Wherever we 
have deductive reasoning, we have mathematics ; but the principles of 
deduction, the recognition of indefinable entities, and the distinguishing 
between such entities, are the business of philosophy. Philosophy is, in 
fact, mainly a question of insight and perception. Entities which are 
perceived by the so-called senses, such as colours and sounds, are, for 
some reason, not commonly regarded as coming within the scope of 
philosophy, except as regards the more abstract of their relations ; but 
it seems highly doubtful whether any such exclusion can be maintained. 
I^ any case, however, since the present work is essentially unconcerned 
with sensible objects, we may confine our remarks to entities which are 
not regarded as existing in space and time. Such entities, if we are to 
know anything about them, must be also in some sense perceived, and 
niust be distinguished one from another; their relations also must be 
m part immediately apprehended. A certain body of indefinable entities 
®^d indemonstrable propositions must form the starting-point for any 
niathematical reasoning ; and it is this starting-point that concerns the 
philosopher. When the philosopher's work has been perfectly accom- 
plished, its results can be wholly embodied in premisses from which 
Reduction may proceed. Now it follows from the very nature of such 
enquiries that results may be disproved, but can never be proved. The 
^^proof will consist in pointing out contradictions and inconsistencies ; 

^- 9 



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130 Number [chap, xv 

but the absence of these can never amount to proof. All depends, in 
the end, upon immediate perception ; and philosophical argument, 
strictly speaking, consists mainly of an endeavour to cause the reader to 
perceive what has been perceived by the author. The argument, in 
short, is not of the nature of proof, but of exhortation. Thus the 
question of the present chapter: Is there any indefinable set of entities 
commonly called numbers, and different from the set of entities above 
defined ? is an essentially philosophical question, to be settled by in- 
spection rather than by accurate chains of reasoning. 

125. In the present chapter, we shall examine the question whether 
the above definition of cardinal numbers in any way presupposes some 
more fundamental sense of number. There are several ways in which 
this may be supposed to be the case. In the first place, the individuals 
which compose classes seem to be each in some sense one^ and it might 
be thought that a one-one relation could not be defined without in- 
troducing the number 1. In the second place, it may very well be 
questioned whether a class which has only one term can be distinguished 
from that one term. And in the third place, it may be held that the 
notion of class presupposes number in a sense different from that above 
defined : it may be maintained that classes arise from the addition of 
individuals, as indicated by the word and^ and that the logical addition 
of classes is subsequent to this addition of individuals. These questions 
demand a new inquiry into the meaning of one and of class ^ and here, 
I hope, we shall find ourselves aided by the theories set forth in Part I. 

As regards the fact that any individual or term is in some sense oiie^ 
this is of course undeniable. But it does not follow that the notion of 
one is presupposed when individuals are spoken of: it may be, on the 
contrary, that the notion of term or individual is the fundamental one, 
from which that of one is derived. This view was adopted in Part I, 
and there seems no reason to reject it. And as for one-one relations, 
they are defined by means of identity, without any mention of OTie^ as 
follows : i? is a one-one relation if, when x and oo' have the relation J? to 
«/, and X has the relation R to y and y\ then x and x are identical, and 
so are y and y' , It is true that here x^ «/, x\ y are each one term, but 
this is not (it would seem) in any way presupposed in the definition. 
This disposes (pending a new inquiry into the nature of classes) of the 
first of the above objections. 

The next question is as to the distinction between a class containing 
only one member, and the one member which it contains. If we could 
identify a class with its defining predicate or class-concept, no difficulty 
would arise on this point. When a certain predicate attaches to one 
and only one term, it is plain that that term is not identical with the 
predicate in question. But if two predicates attach to precisely the 
same terms, we should say that, although the predicates are different, 
the classes which they define are identical, %x. there is only one class 



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124-126] Addition of Terms and Addition of Classes 131 

which both define. If, for example, all featherless bipeds are men, and 
all men are featherless bipeds, the classes men and featherless bipeds are 
identical, though man differs from featherless biped. This shows that a 
class cannot be identified with its class-concept or defining predicate. 
There might seem to be nothing left except the actual terms, so that 
when there is only one term, that term would have to be identical with 
the class. Yet for many formal reasons this view cannot give the 
meaning of the symbols which stand for classes in symbolic logic. For 
example, consider the class of numbers which, added to 3, give 5. This 
is a class containing no terms except the number %. But we can say 
that 2 is a member of this class, i.e, it has to the class that peculiar 
indefinable relation which terms have to the classes they belong to. 
This seems to indicate that the class is different from the one term. 
The point is a prominent one in Peano's Symbolic Logic, and is con- 
nected with his distinction between the relation of an individual to its 
class and the relation of a class to another in which it is contained. 
Thus the class of numbers which, added to 3, give 5, is contained in the 
class of numbers, but is not a number; whereas 2 is a number, but is 
not a class contained in the class of numbers. To identify the two 
relations which Peano distinguishes is to cause havoc in the theory of 
infinity, and to destroy the formal precision of many arguments and 
definitions. It seems, in fact, indubitable that Peano's distinction is 
just, and that some way must be found of discriminating a term from 
a class containing that term only. 

126. In order to decide this point, it is necessary to pass to our 
third difficulty, and reconsider the notion of class liseli. This notion 
appears to be connected with the notion of denoting^ explained in Part I, 
Chapter v. We there pointed out five ways of denoting, one of which 
we called the numerical conjunction. This was the kind indicated by all. 
This kind of conjunction appears to be that which is relevant in the 
case of classes. For example, man being the class-concept, all men will 
be the class. But it will not be all men qua concept which will be the 
class, but what this concept denotes, i.e. certain terms combined in the 
particular way indicated by all. The way of combination is essential, 
since any man or some man is plainly not the class, though either denotes 
combinations of precisely the same terms. It might seem as though, if 
we identify a class with the numerical conjunction of its terms, we must 
deny the distinction of a term from a class whose only member is that 
term. But we found in Chapter x that a class must be always an object 
of a different logical type from its members, and that, in order to avoid 
the proposition xex^ this doctrine must be extended even to classes 
^vhich have only one member. How far this forbids us to identify 
classes with numerical conjunctions, I do not profess to decide ; in any 
case, the distinction between a term and the class whose only member 
it is must be made, and yet classes must be taken extensionally to the 

9—2 



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1 32 Number [chap, x 

degree involved in their being determinate when their members ar 
given. Such classes are called by Frege Werthverldufe; and cardina 
numbers are to be regarded as classes in this sense. 

127. There is still, however, a certain difficulty, which is this: 
class seems to be not many terms, but to be itself a single term, evei 
when many terms are members of the class. This difficulty would seen 
to indicate that the class cannot be identified with all its membei^, bu 
is rather to be regarded as the whole which they compose. In ordei 
however, to state the difficulty in an unobjectionable manner, we mus 
exclude unity and plurality from the statement of it, since these notion 
were to be defined by means of the notion of class. And here it may b 
well to clear up a point which is likely to occur to the reader. Is th 
notion of one presupposed every time we speak of a term ? A term 
it may be said, means one term, and thus no statement can be mad( 
concerning a term without presupposing on£. In some sense of one^ thi 
proposition seems indubitable. Whatever is, is one : being and one, a 
Leibniz remarks, are convertible terms*. It is difficult to be sure hov 
far such statements are merely grammatical. For although whatevei 
is, is one, yet it is equally true that whatever are, are many. But th< 
truth seems to be that the kind of object which is a class, i.e. the kinc 
of object denoted by all men^ or by any concept of a class, is not om 
except where the class has only one term, and must not be made a single 
logical subject. There is, as we said in Part I, Chapter vi, in simple cases ar 
associated single term which is the class as a whole ; but this is sometimes 
absent, and is in any case not identical with the class as many. But ir 
this view there is not a contradiction, as in the theory that verbs anc 
adjectives cannot be made subjects ; for assertions can be made aboui 
classes as many, but the subject of such assertions is many, not one onlj 
as in other assertions. " Brown and Jones are two of Miss Smith's 
suitors "" is an assertion about the class " Brown and Jones," but nol 
about this class considered as a single term. Thus one-ness belongs, ir 
this view, to a certain type of logical subjects, but classes which are nol 
one may yet have assertions made about them. Hence we conclude that 
one-ness is implied, but not presupposed, in statements about a term, 
and " a term '"' is to be regarded as an indefinable. 

128. It seems necessary, however, to make a distinction as regards 
the use of one. The sense in which every object, is otz^, which is 
apparently involved in speaking of an object, is, as Frege urges f, a very 
shadowy sense, since it is applicable to everything alike. But the sense 
in which a class may be said to have one member is quite precise. 
A class u has one member when u is not null, and "x and y are ii^'''' 
implies " x is identical with t/." Here the one-ness is a property of the 

* Ed. Gerhardt, ii, p. 300. 

t Grundlagen der Arithmetih, Breslau^ 1884^ p. ^0. 



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126-130] Addition of Terms and Addition of Classes 133 

class, which may therefore be called a unit-class. The x which is its 
only member may be itself a class of many terms, and this shows that 
the sense of one involved in one term or a term is not relevant to 
Arithmetic, for many terms as such may be a single member of a class 
of classes. One^ therefore, is not to be asserted of terms, but of classes 
having one nlember in the above-defined sense ; 'i.e, " u is one,"" or better 
" li is a unit '' means " u is not null, and ' x and y are iC^ ** implies ' x 
and y are identical \'" The member of Z6, in this case, will itself be none 
or one or many if u is a class of classes ; but if u is a class of terms, 
the member of u will be neither none nor one nor many, but simply 
a term. 

129. The commonly received view, as regards finite numbers, is that 
they result from counting, or, as some philosophers would prefer to 
say, from synthesizing. Unfortunately, those who hold this view have 
not analyzed the notion of counting : if they had done so, they would 
have seen that it is very complex, and presupposes the very numbers 
which it is supposed to generate. 

The process of counting has, of course, a psychological aspect, but 
this is quite irrelevant to the theory of Arithmetic. What I wish now 
to point out is the logical process involved in the act of counting, which 
is as follows. When we say one, two, three, etc., we are necessarily 
considering; some one-one relation which holds between the numbers used 
in counting and the objects counted. What is meant by the " one, two, 
three "" is that the objects indicated by these numbers are their correlates 
with respect to the relation which we have in mind. (This relation, by 
the way, is usually extremely complex, and is apt to involve a reference 
to our state of mind at the moment.) Thus we correlate a class of objects 
with a class of numbers ; and the class of numbers consists of all the 
numbers from 1 up to some number n. The only immediate inference to be 
drawn from this correlation is, that the number of objects is the same as 
the number of numbers from 1 up to n, A further process is required to 
show that this number of numbers is n^ w^hich is only true, as a matter 
of fact, when n is finite, or, in a certain wider sense, when n is olq (the 
smallest of infinite numbers). Moreover the process of counting gives us 
no indication as to what the numbers are, as to why they form a series, 
or as to how it is to be proved (in the cases where it is true) that there 
are n numbers from 1 up to n. Hence counting is irrelevant in the 
foundations of Arithmetic ; and with this conclusion, it may be dismissed 
until we come to order and ordinal numbers. 

130. Let us return to the notion of the numerical conjunction. It 
is plain that it is of such objects as "A and B^' "A and B and C," 
that numbers other than one are to be asserted. We examined such 
objects, in Part I, in relation to classes, with which we found them to 
be identical. Now we must investigate their relation to numbers and 
plurality. 



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134 Number [chap, xv 

The notion to be now examined is the notion of a numerical 
conjunction or, more shortly, a collection. This is not to be identified 
to begin with, with the notion of a class, but is to receive a new and 
independent treatment. By a collection I mean what is conveyed by 
" A and ^'' or "A and B and C,"** or any other enumeration of definite 
terms. The collection is defined by the actual mention of the terms, 
and the terms are connected by and. It would seem that and represents 
a fundamental way of combining terms, and it might be urged that 
just this way of combination is essential if anything is to result of which 
a' number other than 1 is to be asserted. Collections do not presuppose 
numbers, since they result simply from the terms together with and: 
they could only presuppose numbers in the particular case where the 
terms of the collection themselves presupposed numbers. There is a 
grammatical difficulty which, since no method exists of avoiding it, 
must be pointed out and allowed for. A collection, grammatically, is 
one, whereas A and B, or A and B and C, are essentially many. The 
strict meaning of collection is the whole composed of many, but since a 
word is needed to denote the many themselves, I choose to use the word 
collection in this sense, so that a collection, according to the usage here 
adopted, is many and not one. 

As regards what is meant by the combination indicated by and^ it 
gives what we called before the numerical conjunction. That is A and 
B is what is denoted by the concept of a class of which A and B are 
the only terms, and is precisely A and B denoted in the way which is 
indicated by all. We may say, if u be the class-concept corresponding 
to a class of which A and B are the only terms, that "all 2/''s" is a 
concept which denotes the terms AyB combined in a certain way, and 
A and B are those terms combined in precisely that way. Thus A and 
B appears indistinguishable from the class, though distinguishable from 
the class -concept and from the concept of the class. Hence it u he & 
class of more than one term, it seems necessary to hold that u is not 
one, but many, since u is distinguished both from the class-concept and 
from the whole composed of the terms of ii*. Thus we are brought back 
to the dependence of numbers upon classes; and where it is not said 
that the classes in question are finite, it is practically necessary to begin 
with class-concepts and the theory of denoting, not with the theory of 
and which has just been given. The theory of and applies practically 
only to finite numbers, and gives to finite numbers a position which is 
different, at least psychologically, from that of infinite numbers. There 



* A conclusive reason against identifying a class with the whole composed of its 
terms is^ that one of these terms may be the class itself, as in the case ^*^ class is a 
class," or rather '^'^ classes are one among classes," Tlie logical type of the class class 
is of an infinite order^ and therefore the usual objection to ^^ xejc " does not apply in 
this case. 



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130-132] Addition of Terms and Addition of Classes 135 

are, in short, two ways of defining particular finite classes, but there is 
only one practicable way of defining particular infinite classes, namely 
by intension. It is largely the habit of considering classes primarily 
from the side of extension which has hitherto stood in the way of a 
correct logical theory of infinity. 

131. Addition, it should be carefully observed, is not primarily a 
method of forming numbers, but of forming classes or collections. If 
we add iS to J, we do not obtain the number 2, but we obtain A and J?, 
which is a collection of two terms, or a couple. And a couple is defined 
as follows : z/ is a couple if u has terms, and if, if ^ be a term of ?/, there 
is a term of u different from ^, but if x^ y be different terms of w, and z 
differs from x and from y^ then every class to which z belongs differs 
from u. In this definition, only diversity occurs, together with the 
notion of a class having terms. It might no doubt be objected that we 
have to take just two terms x^ y in the above definition : but as a 
matter of fact any finite number can be defined by induction without 
introducing more than one term. For, if n has been defined, a class u 
has 71 -f- 1 terms when, if ^r be a term of w, the number of terms of u 
which differ from x is n. And the notion of the arithmetical sum n + 1 
is obtained from that of the logical sum of a class of n terms and a class 
of one term. When we say 1 + 1 = 2, it is not possible that we should 
mean 1 and 1, since there is only one 1 : if we take 1 as an individual, 
1 and 1 is nonsense, while if we take it as a class, the rule of Symbolic 
Logic applies, according to which 1 and 1 is 1. Thus in the corresponding 
logical proposition, we have on the left-hand side terms of which 1 can 
be asserted, and on the right-hand side we have a couple. That is, 
14-1 = 2 means " one term and one term are two terms," or, stating the 
proposition in terms of variables, "if u has one term and v has one 
term, and u differs from t^, their logical sum has two terms." It is to be 
observed that on the left-hand side w^e have a numerical conjunction of 
propositions, while on the right-hand side we have a proposition con- 
cerning a numerical conjunction of terms. But the true premiss, in the 
above proposition, is not the conjunction of the three propositions, but 
their logical product. This point, however, has little importance in the 
present connection. 

132. Thus the only point which remains is this : Does the notion 
of a term presuppose the notion of 1 't For we have seen that all 
numbers except involve in their definitions the notion of a term, and 
if this in turn involves 1, the definition of 1 becomes circular, and 1 will 
have to be allowed to be indefinable. This objection to our procedure 
is answered by^the doctrine of § 128, that a term is not one in the sense 
which is relevant to Arithmetic, or in the sense which is opposed to 
many. The notion of any term is a logical indefinable, presupposed in 
formal truth and in the whole theory of the variable ; but this notion is 
that of the variable conjunction of terms, which in no way involves the 



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136 Number [chap, xv 

number 1. There is therefore nothing circular in defining the number 1 
by means of the notion of a term or of any term. 

To sum up : Numbers are classes of classes, namely of all classes 
similar to a given class. Here classes have to be understood in the 
sense of numerical conjunctions in the case of classes having many 
terms ; but a class may have no terms, and a class of one term is distinct 
from that term, so that a class is not simply the sum of its terms. Only 
classes have numbers ; of what is commonly called one object, it is not 
true, at least in the sense required, to say that it is one, as appears from 
the fact that the object may be a class of many terms. '^ One object '** 
seems to mean merely " a logical subject in some proposition.'*' Finite 
numbers are not to be regarded as generated by counting, which on the 
contrary presupposes thibm ; and addition is primarily logical addition, 
first of propositions, then of classes, from which latter arithmetical 
addition is derivative. The assertion of numbers depends upon the fact 
that a class of many terms can be a logical subject without being 
arithmetically one. Thus it appeared that no philosophical argument 
could overthrow the mathematical theory of cardinal numbers set forth 
in Chapters xi to xiv. 



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CHAPTER XVI. 



WHOLE AND PART. 

133. For the comprehension of analysis, it is necessary to investigate 
the notion of whole and part, a notion which has been wrapped in 
obscurity — though not without certain more or less valid logical 
reasons — by the writers who may be roughly called Hegelian. In the 
present chapter I shall do my best to set forth a straightforward and 
non-mystical theory of the subject, leaving controversy as far as possible 
on one side. It may be well to point out, to begin with, that I shall 
use the word whole as strictly correlative to part^ so that nothing will 
be called a whole unless it has parts. Simple terms, such as points, 
instants, colours, or the fundamental concepts of logic, will not be called 
wholes. 

Terms which are not classes may be, as we saw in the preceding 
chapter, of two kinds. The first kind are simple : these may be 
characterized, though not defined, by the fact that the propositions 
asserting the being of such terms have no presuppositions. The second 
kind of terms that are not classes, on the other hand, are complex, and 
in their case, their being presupposes the being of certain other terms. 
Whatever is not a class is called a miit^ and thus units are either simple 
or complex. A complex unit is a whole; its parts are other units, 
whether simple or complex, which are presupposed in it. This suggests 
the possibility of defining whole and paii: by means of logical priority, 
a suggestion which, though it must be ultimately rejected, it will be 
necessary to examine at length. 

134. Wherever we have a one-sided formal implication, it may be 
urged, if the two propositional functions involved are obtainable one from 
the other by the variation of a single constituent, then what is implied 
is simpler than what implies it. Thus "Socrates is a man*" implies 
" Socrates is a mortal,"" but the latter proposition does not imply the 
former: also the latter proposition is simpler than the former, since 
man is a concept of which mortal forms part. Again, if we take 
a proposition asserting a relation of two entities A and fi, this 
proposition implies the being of A and the being of J5, and the being of 



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138 Number [chap, xvi 

the relation, none of which implies the proposition, and each of which is 
simpler than the proposition. There will only be equal complexity — 
according to the theory that intension and extension vary inversely as 
one another — in cases of mutual implication, such as "^ is greater 
than J^'' and "j5 is less than J.'^ Thus we might be tempted to set up 
the following definition : A is said to be part of B when B is implies 
J is, but J is does^not imply B is. If this definition could be main- 
tained, whole and part would not be a new indefinable, but would be 
derivative from logical priority. There are, however, reasons why such 
an opinion is untenable. 

The first objection is, that logical priority is not a simple relation : 
implication is simple, but logical priority oi J 1p B requires not only 
'^B implies ^," but also ''A does not imply J^.*" (For convenience, 
I shall say that A implies B when A is implies B is.) This state of 
things, it is true, is realized when A is part of B ; but it seems necessary 
to regard the relation of whole to part as something simple, which must 
be different from any possible relation of one whole to another which is 
not part of it. This would not result from the above definition. For 
example, "A is greater and .betl:er than 5"" implies "B is less than ^," 
but the converse implication does not hold : yet the latter proposition is 
not part of the former*. 

Another objection is derived from such cases as redness and colour. 
These two concepts appear to be equally simple: there is no specification, 
other and simpler than redness itself, which can be added to colour to 
produce redness, in the way in which specifications will turn mortal into 
man. Hence A is red is no more complex than A i^ coloured, although 
there is here a one-sided implication. Redness, in fact, appears to be 
(when taken to mean one particular shade) a simple concept, which, 
although it implies colour, does not contain colour as a constituent. 
The inverse relation of extension and intension, therefore, does not hold 
in all cases. For these reasons, we must reject, in spite of their very 
close connection, the attempt to define whole and part by means of 
implication. 

135. Having failed to define wholes by logical priority, we shall 
not, I think, find it possible to define them at all. The relation of 
whole and part is, it would seem, an indefinable and ultimate relation, 
or rather, it is several relations, often confounded, of which one at least 
is indefinable. The relation of a part to a whole must be differently 
discussed according to the nature both of the whole and of the parts. 
Let us begin with the simplest case, and proceed gradually to those that 
are more elaborate. 

(1) Whenever we have any collection of many terms, in the sense 
explained in the preceding chapter, there the terms, provided there is 

* See Part IV, Chap, xxvii. 



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134, 135] Whole and Part 139 

some non-quadratic propositional function which they all satisfy, together 
form a whole. In the preceding chapter we regarded the class as formed by 
all the terms, but usage seems to show no reason why the class should not 
equally be regarded as the whole composed of all the terms in those cases 
where there is such a whole. The first is the class as many, the second 
the class as one. Each of the terms then has to the whole a certain 
indefinable relation*, which is one meaning of the relation of whole and 
part. The whole is, in this case, a whole of a particular kind, which 
I shall call an aggregate : it differs from wholes of other kinds by the 
fact that it is definite as soon as its constituents are known. 

(2) But the above relation holds only between the aggregate and 
the single terms of the collection composing the aggregate : the relation 
to our aggregate of aggregates containing some but not all the terms 
of our aggregate, is a different relation, though also one which would be 
commonly called a relation of part to whole. For example, the relation 
of the Greek nation to the human race is different from that of Socrates 
to the human race ; and the relation of the whole of the primes to the 
whole of the numbers is different from that of 2 to the whole of the 
numbers. This most vital distinction is due to Peanof. The relation 
of a subordinate aggregate to one in which it is contained can be defined, 
as was explained in Part I, by means of implication and the first kind of 
relation of part to whole. If w, v be two aggregates, and for every 
value of ^ " ^ is a w "" implies " ^r is a z;,"" then, provided the converse 
implication does not hold, u is a proper part (in the second sense) of v. 
This sense of whole and part, therefore, is derivative and definable. 

(S) But there is another kind of whole, which may be called a unity. 
Such a whole is always a proposition, though it need not be an asserted 
proposition. For example, "^ differs from J?,"" or ''^^s difference from 
-B,^ is a complex of which the parts are A and B and difference ; but 
this sense of whole and part is different from the previous senses, since 
''A differs from J?""* is not an aggregate, and has no parts at all in the 
first two senses of parts. It is parts in this third sense that are chiefly 
considered by philosophers, while the first two senses are those usually 
relevant in symbolic logic and mathematics. This third sense of paH is 
the sense which coiTesponds to analysis : it appears to be indefinable, 
like the first sense— i.^., I know no way of defining it. It must be held 
that the three senses are always to be kept distinct : i.e., if A is part 
o{ B in one sense, while B is part of C in another, it must not be 
inferred (in general) that A is part of C in any of the three senses. But 
we may make a fourth general sense, in which anything which is part in 

* Which may^ if we choose^ be taken as Peano's c. The objection to this 
peaning for f is that not every propositional function defines a whole of the kind 
'^©quired. The whole differs from the class as many by being of the same type as its 
terms, 

+ Cf. e.g. f: 1901, § 1, Prop. 4. 4, note (p. 12). 



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140 Nuviher [chap, xvi 

any sense, or part in one sense of part in another, is to be called a part. 
This sense, however, has seldom, if ever, any utility in actual discussion. 

136. The difference between the kinds of wholes is important, 
and illustrates a fundamental point in Logic. I shall therefore repeat 
it in other words. Any collection whatever, if defined by a non-quadratic 
propositional function, though as such it is many, yet composes a whole, 
whose parts are the terms of the collection or any whole composed of some 
of the terms of the collection. It is highly important to realize the difi^er- 
ence between a whole and all its parts, even in this case where the difference 
is a minimum. The word collection^ being singular, applies more strictly 
to the whole than to all the parts ; but convenience of expression has led 
me to neglect grammar, and speak of all the terms as the collection. 
The whole formed of the terms of the collection I call an aggregate. 
Such a whole is completely specified when all its simple constituents, are 
specified ; its parts have no direct connection inter se^ but only the 
indirect connection involved in being parts of one and the same whole. 
But other wholes occur, which contain relations or what may be called 
predicates, not occurring simply as terms in a collection, 43ut as relating 
or qualifying. Such wholes are always propositions. These are not 
completely specified when their parts are all known. Take, as a simple 
instance, the proposition "^ differs from J?,"" where A and B are simple 
terms. The simple parts of this w^hole are A and B and difference ; but 
the enumeration of these three does not specify the whole, since there 
are two other wholes composed of the same parts, namely the aggregate 
formed of A and B and difference, and the proposition "iB differs 
from A!^ In the former case, although the whole was different from 
all its parts, yet it was completely specified by specifying its parts ; but 
in the present case, not only is the whole different, but it is not even 
specified by specifying its parts. We cannot explain this fact by saying 
that the parts stand in certain relations which are omitted in the 
analysis; for in the above case of "^ differs from B^ the relation was 
included in the analysis. The fact seems to be that a relation is one 
thing when it relates, and another when it is merely enumerated as a 
term in a collection. There are certain fundamental difficulties in this 
view, which however I leave aside as irrelevant to our present purpose*. 

Similar remarks apply to A is, which is a whole composed of A and 
Being, but is different from the whole formed of the collection A and 
Being. A is one raises the same point, and so does A and B are two. 
Indeed all propositions raise this point, and we may distinguish them 
among complex terms by the fact that they raise it. 

Thus we see that there are two very different classes of wholes, of 
which the first will be called aggregates, while the second will be called 
unities. ( Unit is a word having a quite different application, since what- 

* See Part I^ Chap, iv, esp. § 54. 



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135-139] Wliole and Part 141 

ever is a class which is not null, and is such that, if x and y be members 
of it, X and y are identical, is a unit.) Each class of wholes consists of 
terms not simply equivalent to all their parts ; but in the case of unities, 
the whole is not even specified by its parts. For example, the parts A^ 
greater than^ J?, may compose simply an aggregate, or either of the 
propositions "A is greater than J?," "B is greater than -i.'" Unities 
thus involve problems from which aggi'egates are free. As aggregates 
are more specially relevant to mathematics than unities, I shall in 
future generally confine myself to the former. 

137. It is important to realize that a whole is a new single term, 
distinct from each of its parts and from all of them : it is one, not many *, 
and is related to the parts, but has a being distinct from theirs. The 
reader may perhaps be inclined to doubt whether there is any need of 
wholes other than unities; but the following reasons seem to make 
aggregates logically unavoidable. (1) We speak of one collection, one 
manifold, etc., and it would seem that in all these cases there really is 
something that is a single term. (2) The theory of fractions, as we shall 
shortly see, appears to depend partly upon aggregates. (3) We shall find 
it necessary, in the theory of extensive quantity, to assume that aggregates, 
even when they are infinite, have what may be called magnitude of 
divisibility, and that two infinite aggregates may have the same number 
of terms without having the same magnitude of divisibility: this theory, 
we shall find, is indispensable in metrical geometry. For these reasons, 
it would seem, the aggregate must be admitted as an entity distinct 
from all its constituents, and having to each of them a certain ultimate 
and indefinable relation. 

138. I have already touched on a very important logical doctrine, 
which the theory of whole and part brings into prominence — I mean the 
doctrine that analysis is falsification. AVhatever can be analyzed is a 
whole, and we have already seen that analysis of wholes is in some 
measure falsification. But it is important to realize the very narrow 
limits of this doctrine. We cannot conclude that the parts of a whole 
are not really its parts, nor that the parts are not presupposed in the 
whole in a sense in which the whole is not presupposed in the parts, nor 
yet that the logically prior is not usually simpler than the logically 
subsequent. In short, though analysis gives us the truth, and nothing 
but the truth, yet it can never give us the whole truth. This is the 
only sense in which the doctrine is to be accepted. In any wider sense, 
it becomes merely a cloak for laziness, by giving an excuse to those who 
dislike the labour of analysis. 

139. It is to be observed that what we called classes as one may 
always, except where they contain one term or none, or are defined by 
quadratic propositional functions, be interpreted as aggregates. The 

■^ /. e. it is of the same logical type as its simple parts. 



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142 Number j[cHAP. xvi 

logical product of two classes as one will be the common part (in the 
second of our three senses) of the two aggregates, and their sum will 
be the aggregate which is identical with or part of (again in the second 
sense) any aggregate of which the two given aggregates are parts, but is 
neither identical with nor part of any other aggregate*. The relation 
of whole and part, in the second of our three senses, is transitive and 
asymmetrical, but is distinguished from other such relations by the fact 
of allowing logical addition and multiplication. It is this peculiarity 
which forms the basis of the Logical Calculus as developed by writers 
previous to Peano and Frege (including Schroder) +. But wherever infinite 
wholes are concerned it is necessary, and in many other cases it is 
practically unavoidable, to begin wqth a class-concept or predicate or 
propositional function, and obtain the aggregate from this. Thus the 
theory of whole and part is less fundamental logically than that of 
predicates or class-concepts or propositional functions ; and it is for 
this reason that the consideration of it has been postponed to so late 
a stage. 

^ Cf. Peano, F, 1901, § 2, Prop. 1*0 (p. 19). 

t See e.g. his Algebra der Logik, Vol. i (Leipzig, 1890). 



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CHAPTER XVII. 



INFINITE WHOLES. 



140. In the present chapter the special difficulties of infinity are 
not to be considered : all these are postponed to Part V. My object 
now is to consider two questions: (1) Are there any infinite wholes? 
(2) If so, must an infinite whole which contains parts in the second of 
our three senses be an aggregate of parts in the first sense ? In order to 
avoid the reference to the first, second and third senses, I propose hence- 
forward to use the following phraseology : A part in the first sense is to 
be called a te7yn of the whole* ; a part in the second sense is to be called 
a pa?^ simply ; and a part in the third sense will be called a constituent 
of the whole. Thus terms and parts belong to aggregates, while con- 
stituents belong to unities. The consideration of aggregates and unities, 
where infinity is concerned, must be separately conducted. I shall begin 
with aggregates. 

An infinite aggregate is an aggregate corresponding to an infinite 
class, i.e. an aggregate which has an infinite number of terms. Such 
aggregates are defined by the fact that they contain parts which have 
as many terms as themselves. Our first question is : Are there any such 
aggregates ? 

Infinite aggregates are often denied. Even Leibniz, favourable as 
he was to the actual infinite, maintained that, where infinite classes are 
concerned, it is possible to make valid statements about ani/ term of the 
class, but not about all the terms, nor yet about the whole which (as he 
would say) they do not compose f. Kant, again, has been much criticised 
for maintaining that space is an infinite given whole. Many maintain 
that every aggregate must have a finite number of terms, and that 
where this condition is not fulfilled there is no true whole. But I do 
not believe that this view ,can be successfully defended. Among those 
who deny that space is a given whole, not a few would admit that what 
they are pleased to call a finite space may be a given whole, for instance, 

* A part in this sense will also be sometimes called a simple or indivisible part. 
t Cf. Phil. Werke, ed. Gerhardt, ii, p. 315 ; also i, p. 338, v, pp. 144-5. 



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144 Number [chap, xvn 

the space in a room, a box, a bag, or a book-case. But such a space is 
only finite in a psychological sense, i.e. in the sense that we can take it 
in at a glance : it is not finite in the sense that it is an aggregate of a 
finite number of terms, nor yet a unity of a finite number of constituents. 
Thus to admit that such a space can be a whole is to admit that there 
are wholes which are not finite. (This does not follow, it should be 
observed, from the admission of material objects apparently occupying 
finite spaces, for it is always possible to hold that such objects, though 
apparently continuous, consist really of a large but finite number of 
material points.) With respect to time, the same argument holds : to 
say, for example, that a certain length of time elapses between sunrise 
and sunset, is to admit an infinite Avhole, or at least a whole which is not 
finite. It is customary with philosophers to deny the reality of space 
and time, and to deny also that, if they were real, they would be 
aggregates. I shall endeavour to show, in Part VI, that these denials 
are supported by a faulty logic, and by the now resolved difficulties of 
infinity. Since science and common sense join in the opposite view, it 
will therefore be accepted ; and thus, since no argument a priori can 
now be adduced against infinite aggregates, we derive from space and 
time an argument in their favour. 

Again, the natural numbers, or the fractions between and 1, or the 
sum-total of all colours, are infinite, and seem to be true aggregates : 
the position that, although true propositions can be made about ani/ 
number, yet there are no true propositions about all numbers, could be 
supported formerly, as Leibniz supported it, by the supposed contra- 
dictions of infinity, but has become, since Cantor's solution of these 
contradictions, a wholly unnecessary paradox. And where a collection 
can be defined by a non-quadratic propositional function, this must be 
held, I think, to imply that there is a genuine aggregate composed 
of the terms of the collection. It may be observed also that, if there 
were no infinite wholes, the word Universe would be wholly destitute of 
meaning. 

141. We must, then, admit infinite aggregates. It remains to ask 
a more difficult question, namely : Are we to admit infinite unities ? 
This question may also be stated in the form : Are there any 
infinitely complex propositions ? This question is one of gi'eat logical 
importance, and we shall require much care both in stating and in 
discussing it. 

The first point is to be clear as to the meaning of an infinite unity. 
A unity will be infinite when the aggregate of all its constituents is 
infinite, but this scarcely constitutes the meaning of an infinite unity. 
In order to obtain the meaning, we must introduce the notion of a 
simple constituent. We may observe, to begin with, that a constituent 
of a constituent is a constituent of the unity, ke, this form of the 
relation of part to whole, like the second, but unlike the first form, is 



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140, 141] Infinite M^oles 145 

transitive. A simple constituent may now be defined as a constituent 
which itself has no constituents. We may assume, in order to eliminate 
the question concerning aggregates, that no constituent of our unity is 
to be an aggregate, or, if there be a constituent which is an aggregate, 
then this constituent is to be taken as simple. (This view of an aggre- 
gate is rendered legitimate by the fact that an aggregate is a single term, 
and does not have that kind of complexity which belongs to propositions.) 
With this the definition of a simple constituent is completed. 

We may now define an infinite unity as follows : A unity is finite 
when, and only when, the aggregate of its simple constituents is flhite. 
In all other cases a unity is said to be infinite. We have to inquire 
whether there are any such unities *. 

If a unity is infinite, it is possible to find a constituent unity, which 
again contains a constituent unity, and so on without end. If there be 
any unities of this nature, two cases are prima fiicie ipossihle. (1) There 
may be simple constituents of our unity, but these must be infinite in 
number. (£) There may be no simple constituents at all, but all 
constituents, without exception, may be complex ; or, to take a slightly 
more complicated case, it may happen that, although there are some 
simple constituents, yet these and the unities composed of them do not 
constitute all the constituents of the original unity. A unity of either 
of these two kinds will be called infinite. The two kinds, though 
distinct, may be considered together. 

An infinite unity will be an infinitely complex proposition : it will 
not be analyzable in any way into a finite number of constituents. It 
thus diffei-s radically from assertions about infinite aggregates. For 
example, the proposition '^ any number has a successor "" is composed of 
a finite number of constituents : the number of concepts entering into it 
can be enumerated, and in addition to these there is an infinite aggregate 
of terms denoted in the way indicated by ani/, which counts as one 
constituent. Indeed it may be said that the logical purpose which is 
served by the theory of denoting is, to enable propositions of finite 
complexity to deal with infinite classes of terms : this object is effected 
by all, any, and every, and if it were not effected, every general pro- 
position about an infinite class would have to be infinitely (complex. 
Now, for my part, I see no possible way of deciding whether propositions 
of infinite complexity are possible or not ; but this at least is clear, that 
all the propositions known to us (and, it would seem, all propositions 
that we can know) are of finite complexity. It is only by obtaining 
such propositions about infinite classes that we are enabled to deal with 
infinity ; and it is a remarkable and fortunate fact that this method is 
successful. Thus the question whether or not there are infinite unities 
must be left unresolved ; the only thing we can say, on this subject, is 

* In Leibniz's philosophy, all contingent things are infinite unities. 
R. 10 



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146 Number [chap, xvh 

that no such unities occur in any department of human knowledge, and 
therefore none such are relevant to the foundations of mathematics. 

142. I come now to our second question : Must an infinite whole 
which contains parts be an aggregate of terms ? It is often held, for 
example, that spaces have parts, and can be divided ad lib.^ but that 
they have no simple parts, i,e. they are not aggregates of points. The 
same view is put forward as regards periods of time. Now it is plain 
that, if our definition of a part by means of terms {i.e. of the second 
sense of part by means of the first) was correct, the present problem can 
never arise, since parts only belong to aggregates. But it may be urged 
that the notion of part ought to be taken as an indefinable, and that 
therefore it may apply to other wholes than aggregates. This will 
require that we should add to aggregates and unities a new kind of 
whole, con-esponding to the second sense of pai^t. This will be a whole 
which has parts in the second sense, but is not an aggregate or a unity. 
Such a whole seems to be what many philosophers are fond of calling a 
continuum, and space and time are often held to aflbrd instances of such 
a whole. 

Now it may be admitted that, among infinite wholes, we find a 
distinction which seems relevant, but which, I believe, is in reality 
merely psychological. In some cases, we feel no doubt as to the terms, 
but great doubt as to the whole, while in others, the whole seems 
obvious, but the terms seem a precarious inference. The ratios between 
and 1, for instance, are certainly indivisible entities; but the whole 
aggregate of ratios between and 1 seems to be of the nature of a 
construction or inference. On the other hand, sensible spaces and times 
seem to be obvious wholes ; but the inference to indivisible points and 
instants is so obscure as to be often regarded as illegitimate. This 
distinction seems, however, to have no logical basis, but to be wholly 
dependent on the nature of our senses. A slight familiarity with co- 
ordinate geometry suffices to make a finite length seem strictly analogous 
to the stretch of fractions between and 1. It must be admitted, 
nevertheless, that in cases where, as with the fractions, the indivisible 
parts are evident on inspection, the problem with which we are con- 
cerned does not arise. But to infer that all infinite wholes have 
indivisible parts merely because this is known to be the case with some 
of them, would certainly be rash. The general problem remains, 
therefore, namely : Given an infinite whole, is there a universal reason 
for supposing that it contains indivisible parts ? 

143. In the first place, the definition of an infinite whole must not 
be held to deny that it has an assignable number of simple parts which 
do not reconstitute it. For example, the stretch of fractions from to 1 
has three simple parts, ^, i, §. But these do not reconstitute the 
whole, that is, the whole has other parts which are not parts of the 
assigned parts or of the sum of the assigned parts. Again, if we form a 



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141-143] Infinite Wholes 147 

whole out of the number 1 and a Ime an inch long, this whole certainly 
has one simple part, namely 1. Such a case as this may be excluded by 
asking whether every part of our whole either is simple or contains 
simple parts. In this case, if our whole be formed by adding n simple 
terms to an infinite whole, the n simple terms can be taken away, and 
the question can be asked concerning the infinite whole which is left. 
But again, the meaning of our question seems hardly to be: Is our 
infinite whole an actual aggregate of innumerable simple parts ? This is 
doubtless an important question, but it is subsequent to the question we 
are asking, which is : Are there always simple parts at all ? We may 
observe that, if a finite number of simple parts be found, and taken 
away from the whole, the remainder is always infinite. For if not, it 
would have a finite number ; and since the term of two finite numbers is 
finite, the original whole would then be finite. Hence if it can be 
shown that every infinite whole contains one simple part, it follows that 
it contains an infinite number of them. For, taking away the one 
simple part, the remainder is an infinite whole, and therefore has a new 
simple part, and so on. It follows that every part of the whole either is 
simple, or contains simple parts, provided that every infinite whole has 
at least one simple part. But it seems as hard to prove this as to prove 
that every infinite whole is an aggregate. 

If an infinite whole be divided into a finite number of parts, one at 
least of these parts must be infinite. If this be again divided, one of its 
parts must be infinite, and so on. Thus no finite number of divisions 
will reduce all the parts to finitude. Successive divisions give an endless 
series of parts, and in such endless series there is (as we shall see in 
Parts IV and V) no manner of contradiction. Thus there is no method 
of proving by actual division that every infinite whole must be an 
aggregate. So far as this method can show, there is no more reason for 
simple constituents of infinite wholes than for a first moment in time or 
a last finite number. 

But perhaps a contradiction may emerge in the present case from the 
connection of whole and part with logical priority. It certainly seems a 
greater paradox to maintain that infinite wholes do not have indivisible 
parts than to maintain that there is no first moment in time or furthest 
limit to space. This might be explained by the fact that we know many 
simple terms, and some infinite wholes undoubtedly composed of simple 
terms, whereas we know of nothing suggesting a beginning of time or 
space. But it may perhaps have a more solid basis in logical priority. 
For the simpler is always implied in the more complex, and therefore 
there can be no truth about the more complex unless there is truth 
about the simpler. Thus in the analysis of our infinite whole, we are 
always dealing with entities which would not be at all unless their 
constituents were. This makes a real difl^erence from the time-series, for 
example : a moment does not logically presuppose a previous moment, 

10—2 



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148 Number [chap, xvn 

and if it did it would perhaps be self-contradictory to deny a first 
moment, as it has been held (for the same reason) self-contradictory to 
deny a First Cause. It seems to follow that infinite wholes would not 
have Being at all, unless there were innumerable simple Beings whose 
Being is presupposed in that of the infinite wholes. For where the 
presupposition is false, the consequence is false also. Thus there seems 
a special reason for completing the infinite regress in the case of infinite 
wholes, which does not exist where other asymmetrical transitive relations 
are concerned. This is another instance of the peculiarity of the relation 
of whole and part : a relation so important and fundamental that almost 
all our philosophy depends upon the theory we adopt in regard to it. 

The same argument may be otherwise stated by asking how our 
infinite wholes are to be defined. The definition must not be infinitely 
complex, since this would require an infinite unity. Now if there is any 
definition which is of finite complexity, this cannot be obtained from 
the parts, since these are either infinitely numerous (in the case of an 
aggregate), or themselves as complex as the whole (in the case of a 
whole which is not an aggregate). But any definition which is of finite 
complexity will necessarily be intensional, i.e, it will give some character- 
istic of a collection of terms. There seems to be no other known method 
of defining an infinite whole, or of obtaining such a whole in a way not 
involving any infinite unity. 

The above argument, it must be admitted, is less conclusive than 
could be wished, considering the great importance of the point at issue. 
It may, however, be urged in support of it that all the arguments on 
the other side depend upon the supposed difficulties of infinity, and are 
therefore wholly fallacious ; also that the procedure of Geometry and 
Dynamics (as will be shown in Parts VI and VII) imperatively demands 
points and instants. In all applications, in short, the results of the 
doctrine here advocated are far simpler, less paradoxical, and more 
logically satisfactory, than those of the opposite view. I shall therefore 
assume, throughout the remainder of this work, that all the infinite 
wholes with which we shall have to deal are aggregates of terms. 



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CHAPTER XVIII. 



RATIOS AND FRACTIONS. 



144. The present chapter, in so far as it deals with relations of 
integers, is essentially confined to finite integers : those that are infinite 
have no relations strictly analogous to what are usually called ratios. 
But I shall distinguish ratios, as relations between integers, from 
fractions, which are relations between aggregates, or rather between 
their magnitudes of divisibility ; and fractions, we shall find, may 
express relations which hold where both aggregates are infinite. It will 
be necessary to begin with the mathematical definition of ratio, before 
proceeding to more general considerations. 

Ratio is commonly associated with multiplication and division, and 
in this way becomes indistinguishable from fractions. But multiplication 
and division are equally applicable to finite and infinite numbers, though 
in the case of infinite numbers they do not have the properties which 
connect them with ratio in the finite case. Hence it becomes desirable 
to develop a theory of ratio which shall be independent of multiplication 
and division. 

Two finite numbers are said to be consecutive when, if 2/ be a class 
having one of the numbers, and one term be added to w, the resulting 
class has the other number. To be consecutive is thus a relation which 
is one-one and asymmetrical. If now a number a has to a number b 
the wth power of this relation of consecutiveness (powers of relations 
being defined by relative multiplication), then we have a -\- n = b. This 
equation expresses, between a and 6, a one-one relation which is deter- 
nainate when n is given. If now the mth power of this relation holds 
between a' and b\ we shall have a' + mn = b\ Also we may define mn as 
+ 7nn. If now we have three numbers a, b, c such that ab = c, this 
equation expresses between a and c a one-one relation which is deter- 
ininate when b is given. Let us call this rfilation B. Suppose we have 
also ab'=c. Then a has to a' a relation which is the relative product 
of B and the converse of B\ where B' is derived from V && B was derived 
from 6. This relation we define as the ratio of a to a. This theory 
bas the advantage that it applies not only to finite integers, but to 



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150 Number [chap, xvm 

all other series of the same type, i.e. all series of type the which I call 
progressions. 

145. The only point which it is important, for our present purpose, 
to observe as regards the above definition of ratios is, that they are 
one-one relations between finite integers, which are with one exception 
asymmetrical, which are such that one and only one holds between any 
specified pair of finite integers, which are definable in terms of consecu- 
tiveness, and which themselves form a series having no first or last term 
and having a term, and therefore an infinite number of terms, between 
any two specified terms. From the fact that ratios are relations it 
results that no ratios are to be identified with integers : the ratio of 2 to 
1, for example, is a wholly different entity from % When, therefore, 
we speak of the series of ratios as containing integers, the integers said 
to be contained are not cardinal numbers, but relations which have a 
certain one-one correspondence with cardinal numbers. The same remark 
applies to positive and negative numbers. The n\h power of the relation 
of consecutiveness is the positive number + n, which is plainly a wholly 
different concept from the cardinal number n. The confusion of entities 
with others to which they have some important one-one relation is an 
error to which mathematicians are very liable, and one which has 
produced the greatest havoc in the philosophy of mathematics. We 
shall find hereafter innumerable other instances of the same error, and it 
is well to realize, as early as possible, that any failure in subtlety of 
distinctions is sure, in this subject at least, to cause the most disastrous 
consequences. 

There is no difficulty in connecting the above theory of ratio with 
the usual theory derived from multiplication and division. But the 
usual theory does not show, as the present theory does, why the infinite 
integers do not have ratios strictly analogous to those of finite integers. 
The fact is, that ratio depends upon consecutiveness, and consecutiveness 
as above defined does not exist among infinite integers, since these are 
unchanged by the addition of 1. 

It should be observed that what is called addition of ratios demands 
a new set of relations among ratios, relations which may be called 
positive and negative ratios, just as certain relations among integers are 
positive and negative integers. This subject, however, need not be 
further developed. 

146. The above theory of ratio has, it must be confessed, a highly 
artificial appearance, and one which makes it seem extraordinary ,that 
ratios should occur in daily life. The fact is, it is not ratios, but 
fractions, that occur, and fractions are not purely arithmetical, but are 
really concerned with relations of whole and part. 

Propositions asserting fractions show an important difference from 
those asserting integers. We can say A is one, A and B are two, and 
so on ; but we cannot say A is one-third, or A and B are two-thirds. 



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144-147] Ratios and Fractions 151 

There is always need of some second entity, to which our first has some 
fractional relation. We say y^ is one-third of C, A and B together are 
two-thirds of C, and so on. Fractions, in short, are either relations of 
a simple part to a whole, or of two wholes to one another. But it 
is not necessary that the one whole, or the simple part, should be part 
of the other whole. In the case of finite wholes, the matter seems 
simple : the fraction expresses the ratio of the number of parts in the 
one to the number in the other. But the consideration of infinite 
wholes will show us that this simple theory is inadequate to the facts. 

147. There is no doubt that the notion of half a league, or half 
a day, is a legitimate notion. It is therefore necessary to find some 
sense for fractions in which they do not essentially depend upon number. 
For, if a given period of twenty-four hours is to be divided into two 
continuous portions, each of which is to be half of the whole period, 
there is only one way of doing this : but Cantor has shown that every 
possible way of dividing the period into two continuous portions divides 
it into two portions having the same number of terms. There must be, 
therefore, some other respect in which two periods of twelve hours are 
equal, while a period of one hour and another of twenty -three hours 
are unequal. I shall have more to say upon this subject in Part III ; 
for the present I will point out that what we want is of the nature of a 
magnitude, and that it must be essentially a property of ordered wholes. 
I shall call this property magnitude of divisibility. To say now that A is 
one-half of B means : jB is a whole, and if B be divided into two similar 
parts which have both the same magnitude of divisibility as each other, 
then A has the same magnitude of divisibility as each of these parts. 
We may interpret the fraction \ somewhat more simply, by regarding 
it as a relation (analogous to ratio so long as finite wholes are concerned) 
between two magnitudes of divisibility. Thus finite integral fractions 
(such as n/1) will measure the relation of the divisibility of an aggregate 
of n terms to the divisibility of a single term ; the converse relation will 
be \\n. Thus here again we have a new class of entities which is in 
danger of being confused with finite cardinal integers, though in reality 
quite distinct. Fractions, as now interpreted, have the advantage (upon 
which all metrical geometry depends) that they introduce a discrimina- 
tion of greater and smaller among infinite aggregates having the same 
number of terms. We shall see more and more, as the logical inadequacy 
of the usual accounts of measurement is brought to light, how absolutely 
essential the notion of magnitude of divisibility really^is. Fractions, 
then, in the sense in which they may express relations of infinite 
aggregates — and this is the sense which they usually have in daily life — 
are really of the nature of relations between magnitudes of divisibility ; 
and magnitudes of divisibility are only measured by number of parts 
where the aggregates concerned are finite. It may also be observed 
(though this remark is anticipatory) that, w.hereas ratios, as above 



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152 Number [chap, xviii 

defined, are essentially rational, fractions, in the sense here given to 
them, are also capable of irrational values. But the development of 
this topic must be left for Part V. 

148. We may now sum up the results obtained in Part II. In the 
first four chapters, the modern mathematical theory of cardinal integers, 
as it results from the joint labours of arithmeticians and symbolic 
logicians, was briefly set forth. Chapter xi explained the notion of 
similar classes, and showed that the usual formal properties of integers 
result from defining them as classes of similar classes. In Chapter xii, 
we showed how arithmetical addition and multiplication both depend 
upon logical addition, and how both may be defined in a way which 
applies equally to finite and infinite numbers, and to finite and infinite 
sums and products, and which moreover introduces nowhere any idea of 
order. In Chapter xiii, we gave the strict definition of an infinite class, 
as one which is similar to a class resulting from taking away one of its 
terms ; and we showed in outline how to connect this definition with the 
definition of finite numbers by mathematical induction. The special 
theory of finite integers was discussed in Chapter xiv, and it was shown 
how the primitive propositions, which Peano proves to be sufficient in 
this subject, can all be deduced from our definition of finite cardinal 
integers. This confirmed us in the opinion that Arithmetic contains no 
indefinables or indemonstrables beyond those of general logic. 

We then advanced, in Chapter xv, to the consideration of philoso- 
phical questions, with a view of testing critically the above mathematical 
deductions. We decided to regard both tervi and a term as indefinable, 
and to define the number 1, as well as all other numbers, by means of these 
indefinables (together with certain others). We also found it necessary 
to distinguish a class from its class-concept, since one class may have 
several different class-concepts. We decided that a class consists of all 
the terms denoted by the class-concept, denoted in a certain indefinable 
manner ; but it appeared that both common usage and the majority of 
mathematical purposes would allow us to identify a class with the whole 
formed of the terms denoted by the class-concept. The only reasons 
against this view were, the necessity of distinguishing a class containing 
only one term from that one term, and the fact that some classes are 
members of themselves. We found also a distinction between finite and 
infinite classes, that the former can, while the latter cannot, be defined 
extensionally, i.e, fay actual enumeration of their terms. We then 
proceeded to discuss what may be called the addition of individuals, 
i.e, the notion involved in "^ and j^*" ; and we found that a more or less 
independent theory oi finite integers can be based upon this notion. 
But it appeared finally, in virtue of our analysis of the notion of dass^ 
that this theory was really indistinguishable from the theory previously 
expounded, the only difference being that it adopted an extensional 
definition of classes. 



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147, 148] Ratios and Fractions 153 

Chapter xvi dealt with the relation of whole and part. We found 
that there are two indefinable senses of this relation, and one definable 
sense, and that there are two correspondingly different sorts of wholes, 
which we called unities and aggregates respectively. We saw also that, 
by extending the notion of aggregates to single terms and to the null- 
class, we could regard the whole of the traditional calculus of Symbolic 
Logic as an algebra specially applicable to the relations of wholes and 
parts in the definable sense. We considered next, in Chapter xvii, the 
notion of an infinite whole. It appeared that infinite unities, even if 
they be logically possible, at any rate never appear in anything accessible 
to human knowledge. But infinite aggregates, we found, must be ad- 
mitted ; and it seemed that all infinite wholes which are not unities 
must be aggregates of terms, though it is by no means necessary that the 
terms should be simple. (They must, however, owing to the exclusion 
of infinite unities, be assumed to be oi finite complexity.) 

In Chapter xviii, finally, we considered ratios and fractions : the former 
were found to be somewhat complicated relations of finite integers, while 
the latter were relations between the divisibilities of aggregates. These 
divisibilities being magnitudes, their further discussion belongs to Part III, 
in which the general nature of quantity is to be considered. 



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PART III. 

QUANTITY. 



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CHAPTER XIX. 



THE MEANING OF MAGNITUDE. 

149. Among the traditional problems of mathematical philosophy, 
few are more important than the relation of quantity to number. 
Opinion as to this relation has undergone many revolutions. Euclid, 
as is evident from his definitions of ratio and proportion, and indeed 
from his whole procedure, was not persuaded of the applicability of 
numbers to spatial magnitudes. When Des Cartes and Vieta, by the 
introduction of co-ordinate Geometry, made this applicability a funda- 
mental postulate of their systems, a new method was founded, which, 
however fruitful of results, involved, like most mathematical advances of 
the seventeenth century, a diminution of logical precision and a loss in 
subtlety of distinction. What was meant by measurement, and whether 
all spatial magnitudes were susceptible of a numerical measure, were 
questions for whose decision, until very lately, the necessary mathe- 
matical instrument was lacking; and even now much remains to be 
done before a complete answer can be given. The view prevailed that 
number and quantity were the objects of mathematical investigation, 
and that the two were so similar as not to require careful separation. 
Thus number was applied to quantities without any hesitation, and 
conversely, where existing numbers were found inadequate to measure- 
ment, new ones were created on the sole ground that every quantity 
must have a numerical measure. 

All this is now happily changed. Two different lines of argument, 
conducted in the main by different men, have laid the foundations both 
for large generalizations, and for thorough accuracy in detail. On the 
one hand, Weierstrass, Dedekind, Cantor, and their followers, have 
pointed out that, if irrational numbers are to be significantly employed as 
measures of quantitative fractions, they must be defined without reference 
to quantity ; and the same men who showed the necessity of such a 
definition have supplied the want which they had created. In this way, 
during the last thirty or forty years, a new subject, which has added 
quite immeasurably to theoretical correctness, has been created, which 
may legitimately be called Arithmetic ; for, starting with integers, it 



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158 Quantity [chap, xix 

succeeds in defining whatever else it requires — rationals, limits, ir- 
rationals, continuity, and so on. It results that, for all Algebra and 
Analysis, it is unnecessary to assume any material beyond the integers, 
which, as we have seen, can themselves be defined in logical terms. 
It is this science, far more than non-Euclidean Geometry, that is really 
fatal to the , Kantian theory of a priori intuitions as the basis of 
mathematics. Continuity and irrationals were formerly the strongholds 
of the school who may be called intuitionists, but these strongholds are 
theirs no longer. Arithmetic has grown so as to include all that can 
strictly be called pure in the traditional mathematics. 

150. But, concurrently with this purist's reform, an opposite advance 
has been effected. New branches of mathematics, which deal neither 
with number nor with quantity, have been invented ; such are the 
Logical Calculus, Projective Geometry, and — in its essence — the Theory 
of Groups. Moreover it has appeared that measurement — if this means 
the correlation, with numbers, of entities which are not numbers or 
aggregates — is not a prerogative of quantities : some quantities cannot 
be measured, and some things which are not quantities (for example 
anharmonic ratios protectively defined) can be measured. Measurement, 
in fact, as we shall see, is applicable to all series of a certain kind — a kind 
which excludes some quantities and includes some things which are 
not quantities. The separation between number and quantity is thus 
complete : each is wholly independent of the other. Quantity, moreover, 
has lost the mathematical importance which it used to possess, owing to 
the fact that most theorems concerning it can be generalized so as to 
become theorems concerning order. It would therefore be natural 
to discuss order before quantity. As all propositions concerning order 
can, however, be established independently for particular instances of 
order, and as quantity will afford an illustration, requiring slightly less 
effort of abstraction, of the principles to be applied to series in general ; 
as, further, the theory of distance, which forms a part of the theory of 
order, presupposes somewhat controversial opinions as to the nature 
of quantity, I shall follow the more traditional course, and consider 
quantity first. My aim will be to give, in the present chapter, a theory 
of quantity which does not depend upon number, and then to show the 
peculiar relation to number which is possessed by two special classes of 
quantities, upon which depends the measurement of quantities wherever 
this is possible. The whole of this Part, however — and it is important 
to realize this — is a concession to tradition ; for quantity, we shall find, 
is not definable in terms of logical constants, and is not properly a 
notion belonging to pure mathematics at all. I shall discuss quantity 
because it is traditionally supposed to occur in mathematics, and because 
a thorough discussion is required for disproving this supposition ; but 
if the supposition did not exist, I should avoid all mention of any such 
notion as quantity. 



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149-152] The meaning of Magnitude 159 

151. In fixing the meaning of such a term as quantity or magnitude^ 
one is faced with the difficulty that, however one may define the word, 
one must appear to depart from usage. This difficulty arises wherever 
two characteristics have been commonly supposed inseparable which, 
upon closer examination, are discovered to be capable of existing apart. 
In the case of magnitude, the usual meaning appears to imply (1) a 
capacity for the relations of greater and less^ {%) divisibility. Of these 
characteristics, the first is supposed to imply the second. But as I 
propose to deny the implication, I must either admit that some things 
which are indivisible are magnitudes, or that some things which are 
greater or less than others are not magnitudes. As one of these de- 
partures from usage is unavoidable, I shall choose the former, which 
I believe to be the less serious. A magnitude, then, is to be defined as 
anything which is greater or less than something else. 

It might be thought that equality should be mentioned, along with 
greater and less, in the definition of magnitude. We shall see reason 
to think, however — paradoxical as such a view may appear — that what 
can be greater or less than some term, can never be equal to any term 
whatever, and vice versa. This will require a distinction, whose necessity 
will become more and more evident as we proceed, between the kind of 
terms that can be equal, and the kind that can be greater or less. The 
former I shall call quantities^ the latter magnitudes. An actual foot- 
rule is a quantity: its length is a magnitude. Magnitudes are more 
abstract than quantities : when two quantities are equal, they have the 
same magnitude. The necessity of this abstraction is the first point to 
be established. 

152. Setting aside magnitudes for the moment, let us consider 
quantities. A quantity is anything which is capable of quantitative 
equality to something else. Quantitative equality is to be distinguished 
from other kinds, such as arithmetical or logical equality. All kinds 
of equality have in common the three properties of being reflexive, 
symmetrical, and transitive, i.e. a term which has this relation at all 
has this relation to itself; if A has the relation to jB, B has it to A ; 
if A has it to B, and B to C, A has it to C*. What it is that dis- 
tinguishes quantitative equality from other kinds, and whether this 
kind of equality is analyzable, is a further and more difficult question, 
to which we must now proceed. 

There are, so far as I know, three main views of quantitative 
equality. There is (1) the traditional view, which denies quantity as 

"^ On the independence of these three properties^ see Peano_, Revue de Mathematique, 
viij p. 22, The reflexive property is not strictly necessary ; what is properly necessary 
and what is alone (at first sight at any rate) true of quantitative equality^ is^ that there 
exists at least one pair of terms having the relation in question. It follows then from 
the other two properties that each of these terms has to itself the relation in 
question. 



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160 Quantity [chap, xix 

an independent idea, and asserts that two terms are equal when, and 
only when, they have the same number of parts. (S) There is what may 
be called the relative view of quantity, according to which equal, greater 
and less are all direct relations between quantities. In this view we 
have no need of magnitude, since sameness of magnitude is replaced 
by the symmetrical and transitive relation of equality. (3) There is 
the absolute theory of quantity, in which equality is not a direct relation, 
but is to be analyzed into possession of a common magnitude, i.e. into 
sameness of relation to a third term. In this case there will be a special 
kind of relation of a term to its magnitude ; between two magnitudes 
of the same kind there will be the relation of greater and less ; while 
equal, greater and less will apply to quantities only in virtue of their 
relation to magnitudes. The difference between the second and third 
theories is exactly typical of a difference which arises in the case of many 
other series, and notably in regard to space and time. The decision 
is, therefore, a matter of very considerable importance. 

153. (1) The kind of equality which consists in having the same 
number of parts has been already discussed in Part II. If this be 
indeed the meaning of quantitative equality, then quantity introduces 
no new idea. But it may be shown, I think, that greater and less have 
a wider field than whole and part, and an independent meaning. The 
arguments may be enumerated as follows : (a) We must admit indi- 
visible quantities ; (/3)l where the number of simple parts is infinite, 
there is no generalization of number which will give the recognized 
results as to inequality ; (7) some relations must be allowed to be 
quantitative, and relations are not even conceivably divisible ; (8) even 
where there is divisibility, the axiom that the whole is greater than the 
part must be allowed to be significant, and not a result of definition. 

(a) Some quantities are indivisible. For it is generally admitted 
that some psychical existents, such as pleasure and pain, are quantitative. 
If now equality means sameness in the number of indivisible parts, we 
shall have to regard a pleasure or a pain as consisting of a collection 
of units, all perfectly simple, and not, in any significant sense, equal 
iiiter se ; for the equality of compound pleasures results on this hypothesis, 
solely from the number of simple ones entering into their composition, 
so that equality is formally inapplicable to indivisible pleasures. If, on 
the other hand, we allow pleasures to be infinitely divisible, so that no 
unit we can take is indivisible, then the number of units in any given 
pleasure is w^hoUy arbitrary, and if there is to be any equality of 
pleasures, we shall have to admit that any two units may be significantly 
called equal or unequal*. Hence we shall require for equality some 
meaning other than sameness as to the number of parts. This latter 

■^ I shall never use the word unequal to mean merely not equal, but always to 
mean greate}- or less, i.e. not equal^ though of the same kind of quantities. 



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152, 153] The meaning of Magnitude 161 

theory, however, seems unavoidable. For there is not only no reason 
to regard pleasures as consisting of definite sums of indivisible units, 
but further — as a candid consideration will, I think, convince anyone — 
two pleasures can always be significantly judged equal or unequal. 
However small two pleasures may be, it must always be significant to 
say that they are equal. But on the theory I am combating, the judg- 
ment in question would suddenly cease to be significant when both 
pleasures were indivisible units. Such a view seems wholly unwarrant- 
able, and I cannot believe that it has been consciously held by those* 
who have advocated the premisses from which it follows. 

(/5) Some quantities are infinitely divisible, and in these, whatever 
definition we take of infinite number, equality is not coextensive with 
sameness in the number of parts. In the first place, equality or 
inequality must always be definite : concerning two quantities of the 
same kind, one answer must be right and the other wrong, though it is 
often not in our power to decide the alternative. From this it follows 
that, where quantities consist of an infinite number of parts, if equality 
or inequality is to be reduced to number of parts at all, it must be 
reduced to number of simple parts ; for the number of complex parts 
that may be taken to make up the whole is wholly arbitrary. But 
equality, for example in Geometry, is far narrower than sameness in the 
number of parts. The cardinal number of parts in any two continuous 
portions of space is the same, as we know from Cantor ; even the ordinal 
number or type is the same for any two lengths whatever. Hence if 
there is to be any spatial inequality of the kind to which Geometry and 
common-sense have accustomed us, we must seek some other meaning for 
equality than that obtained from the number of parts. At this point 
I shall be told that the meaning is very obvious : it is obtained from 
superposition. Without trenching too far on discussions which belong 
to a later part, I may observe {a) that superposition applies to matter, 
not to space, {h) that as a criterion of equality, it presupposes that the 
matter superposed is rigid, {c) that rigidity means constancy as regards 
metrical properties. This shows that we cannot, wdthout a vicious 
circle, define spatial equality by superposition. Spatial magnitude is, in 
fact, as indefinable as every other kind ; and number of parts, in this case 
as in all others where the number is infinite, is wholly inadequate even 
as a criterion. 

(7) Some relations are quantities. This is suggested by the above 
discussion of spatial magnitudes, where it is very natural to base equality 
upon distances. Although this view, as we shall see hereafter, is not 
wholly adequate, it is yet partly true. There appear to be in certain 
spaces, and there certainly are in some series (for instance that of the 

^ E.g, Mr Bradley, '^'^What do we mean by the Intensity of Psychical States.^" 
Mind, N. S. Vol. iv ; see esp. p. 5. 

R. 11 



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162 Quantity [ohap. xix 

rational numbers), quantitative relations of distance among the various 
terms. Also similarity and difference appear to be quantities. Consider 
for example two shades of colour. It seems undeniable that two shades 
of red are more similar to each other than either is to a shade of blue ; 
yet there is no common property in the one case which is not found in 
the other also. Red is a mere collective name for a certain series of 
shades, and the only reason for giving a collective name to this series 
lies in the close resemblance between its terms. Hence red must not be 
regarded as a common property in virtue of which two shades of red 
resemble each other. And since relations are not even conceivably 
divisible, greater and less among relations cannot depend upon number 
of parts. 

(S) Finally, it is well to consider directly the meanings of greater 
and less on the one hand, and of whole and part on the other. Euclid's 
axiom, that the whole is greater than the part, seems undeniably signi- 
ficant ; but on the traditional view of quantity, this axiom would be 
a mere tautology. This point is again connected with the question 
whether superposition is to be taken as the meaning of equality, or as a 
mere criterion. On the latter view, the axiom must be significant, and 
we cannot identify magnitude with number of parts*. 

154. (2) There is therefore in quantity something over and above 
the ideas which we have hitherto discussed. It remains to decide between 
the relative and absolute theories of magnitude. 

The relative theory regards equal quantities as not possessed of any 
common property over and above that of unequal quantities, but as 
distinguished merely by the mutual relation of equality. There is no 
such thing as a magnitude, shared by equal quantities. We must not 
say : This and that are both a yard long ; we must say : This and that 
are equal, or are both equal to the standard yard in the Exchequer. 
Inequality is also a direct relation between quantities, not between 
magnitudes. There is nothing by which a set of equal quantities are 
distinguished from one which is not equal to them, except the relation of 
equality itself. The course of definition is, therefore, as follows : We 
have first a quality or relation, say pleasure, of which there are various 
instances, specialized, in the case of a quality, by temporal or spatio- 
temporal position, and in the case of a relation, by the terms between 
which it holds. Let us, to fi^ ideas, consider quantities of pleasure. 
Quantities of pleasure consist merely of the complexes pleasure at such a 
time^ and pleasure at such another time (to which place may be added, if 
it be thought that pleasures have position in space). In the analysis of 
a particular pleasure, there is, according to the relational theory, no 
other element to be found. But on comparing these particular pleasures, 

* Compare_, with the above discussion_, Meinong*^ Ueber die Bedeutung des Weber - 
schsn Gesetzes, Hamburg and Leipzig^ 1896 ; especially Chap, i^ § 3. 



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153, 154] The meaning of Magnitude 163 

we find that any two have one and only one of three relations, equal, 
greater, and less. Why some have one relation, some another, is a 
question to which it is theoretically and strictly impossible to give an 
answer ; for there is, ex hypothesis no point of difference except temporal 
or spatio-temporal position, which is obviously irrelevant. Equal quan- 
tities of pleasure do not agree in any respect in which unequal ones 
differ: it merely happens that some have one relation and some another. 
This state of things, it must be admitted, is curious, and it becomes 
still more so when we examine the indemonstrable axioms which the 
relational theory obliges us to assume. They are the following (^, jB, C 
being all quantities of one kind) : 

(a) A =Bs or A is greater than jB, or A is less than B, 

(b) A being given, there is always a B^ which may be identical 
with A , such that A =B. 

(c) liA=B,thenB = A. 

(d) If J = ^ and J5 = C, then A = C. 

(e) If A is greater than B^ then B is less than A. 

(f) If A is greater than jB, and B is greater than C, then A is 
greater than C 

(g) If A is greater than jB, and B=Cs then A is greater than C. 

(h) If A == B, and B is greater than C, then A is greater than C. 
From (b\ (c), and (d) it follows that A=A^. From (e) and (f) it 
follows that, if A is less than jB, and B is less than C, then A is less than 
C ; from (c), {e\ and (h) it follows that, if A is less than jB, and B = Cs 
then A is less than C ; from (c), (e% and (g) it follows that, if J = jB, and 
B is less than C, then A is less than C. (In the place of (b) we may put 
the axiom : If ^ be a quantity, then A=A.) These axioms, it will be 
observed, lead to the conclusion that, in any proposition asserting 
equality, excess, or defect, an equal quantity may be substituted any- 
where without affecting the truth or falsehood of the proposition. 
Further, the proposition A=A is an essential part of the theory. Now 
the first of these facts strongly suggests that what is relevant in quanti- 
tative propositions is not the actual quantity, but some property which 
it shares with other equal quantities. And this suggestion is almost 
demonstrated by the second fact, A =A. For it may be laid down that 
the only unanalyzable symmetrical and transitive relation which a term 
can have to itself is identity, if this be indeed a relation. Hence the 
relation of equality should be analyzable. Now to say that a relation is 
analyzable is to say either that it consists of two or more relations 
between its terms, which is plainly not the case here, or that, when it is 
said to hold between two terms, there is some third term to which both 
are related in ways which, when compounded, give the original relation. 

* This does not follow from (o) and (d) alone/ since they do not assert that A is 
ever equal to B. See Peano^ loc. cit. 

11—2 



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164 Quantity [chap xix 

Thus to assert that A is B''^ grandparent is to assert that there is some 
third person C, who is ^''s son or daughter and B\ father or mother. 
Hence if equahty be analyzable^ two equal terms must both be related to 
some third term ; and since a term may be equal to itself, any two equal 
terms must have the savie relation to the third term in question. But 
to admit this is to admit the absolute theory of magnitude. 

A direct inspection of what we mean when we say that two terms 
are equal or unequal will reinforce the objections to the relational 
theory. It seems preposterous to maintain that equal quantities have 
absolutely nothing in common beyond what is shared by unequal 
quantities. Moreover unequal quantities are not merely different : they 
are different in the specific manner expressed by saying that one is 
greater, the other less. Such a difference seems quite unintelligible 
unless there is some point of difference, where unequal quantities are 
concerned, which is absent where quantities are equal. Thus the rela- 
tional theory, though apparently not absolutely self-contradictory, is 
complicated and paradoxical. Both the complication and the paradox, 
we shall find, are entirely absent in the absolute theory. 

155. (3) In the absolute theory, there is, belonging to a set of 
equal quantities, one definite concept, namely a certain magnitude. 
Magnitudes are distinguished among concepts by the fact that they 
have the relations of greater and less (or at least one of them) to other 
terms, which are therefore also magnitudes. Two magnitudes cannot 
be equal, for equality belongs to quantities, and is defined as possession 
of the same magnitude. Every magnitude is a simple and indefinable 
concept. Not any two magnitudes are one greater and the other less ; 
on the contrary, given any magnitude, those which are greater or less 
than that magnitude form a certain definite class, within which any two 
are one greater and the other less. Such a class is called a hind of 
magnitude. A kind of magnitude may, however, be also defined in 
another way, which has to be connected with the above by an axiom. 
Every magnitude is a magnitude of something — pleasure, distance, area, 
etc. — and has thus a certain specific relation to the something of which 
it is a magnitude. This relation is very peculiar, and appears to be 
incapable of further definition. All magnitudes which have this relation 
to one and the same something {e.g. pleasure) are magnitudes of one 
kind ; and with this definition, it becomes an axiom to say that, of two 
magnitudes of the same kind, one is greater and the other less. 

156. An objection to the above theory may be based on the 
relation of a magnitude to that whose magnitude it is. To fix our 
ideas, let us consider pleasure. A magnitude of pleasure is so much 
pleasure, such and such an intensity of pleasure. It seems difficult to 
regard this, as the absolute theory demands, as a simple idea : there 
seem to be two constituents, pleasure and intensity. Intensity need not 
be intensity of pleasure, and intensity of pleasure is distinct from 



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154-156] The meaning of Magnitude 165 

abstract pleasure. But what we require for the constitution of a certain 
magnitude of pleasure is, not intensity in general, but a certain specific 
intensity ; and a specific intensity cannot be indifferently of pleasure or 
of something else. We cannot first settle how much we will have, and 
then decide whether it is to be pleasure or mass. A specific intensity 
must be of a specific kind. Thus intensity and pleasure are not in- 
dependent and coordinate elements in the definition of a given amount 
of pleasure. There are different kinds of intensity, and different magni- 
tudes in each kind ; but magnitudes in different kinds must be different. 
Thus it seems that the common element, indicated by the term intensity 
or magnitude^ is not any thing intrinsic, that can be discovered by analysis 
of a single term, but is merely the fact of being one term in a relation of 
inequality. Magnitudes are defined by the fact that they have this 
relation, and they do not, so far as the definition shows, agree in any- 
thing else. The class to which they all belong, like the married portion 
of a community, is defined by mutual relations among its terms, not by 
a common relation to some outside term — unless, indeed, inequality 
itself were taken as such a term, which would be merely an unnecessary 
complication. It is necessary to consider what may be called the 
extension or field of a relation, as well as that of a class-concept: and 
magnitude is the class which forms the extension of inequality. Thus 
magnitude of pleasure is complex, because it combines magnitude and 
pleasure; but a particular magnitude of pleasure is not complex, for 
magnitude does not enter into its concept at all. It is only a magnitude 
because it is greater or less than certain other terms; it is only a vasigm' 
tuAe oi pleasure because of a certain relation which it has to pleasure. 
This is more easily understood where the particular magnitude has a 
special name. A yard, for instance, is a magnitude, because it is greater 
than a foot ; it is a magnitude of length, because it is what is called 
« length. Thus all magnitudes are simple concepts, and are classified 
into kinds by their relation to some quality or relation. The quantities 
which are instances of a magnitude are particularized by spatio-temporal 
position or (in the case of relations which are quantities) by the terms 
between which the relation holds. Quantities are not properly greater 
or less, for the relations of greater and less hold between their 
magnitudes, which are distinct from the quantities. 

When this theory is applied in the enumeration of the necessary 
axioms, we find a very notable simplification. The axioms in which 
equality appears have all become demonstrable, and we require only the 
following (jL, J/, N being magnitudes of one kind) : 

{a) No magnitude is greater or less than itself. 

(h) L is greater than Ji" or L is less than M, 

{e) If L is greater than Jf , then M is less than Z. 

{d) If L is greater than M and M is greater than iV, then L is 
greater than N. 



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166 Quantity [chap, xix 

The difficult axiom which we formerly called (&) is avoided, as are the 
other axioms concerning equality; and those that remain are simpler 
than our former set. 

157. The decision between the absolute and relative theories can 
be made at once by appealing to a certain general principle, of very 
wide application, which I propose to call the principle of Abstraction. 
This principle asserts that, whenever a relation, of which there are 
instances, has the two properties of being symmetrical and transitive, 
then the relation in question is not primitive, but is analyzable into 
sameness of relation to some other term ; and that this common relation 
is such that there is only one term at most to which a given term can be 
so related, though many terms may be so related to a given term. 
(That is, the relation is like that of son to father: a man may have 
many sons, but can have only one father.) 

This principle, which we have already met with in connection with 
cardinals, may seem somewhat elaborate. It is, however, capable of 
proof, and is merely a careful statement of a very common assumption. 
It is generally held that all relations are analyzable into identity or 
diversity of content. Though I entirely reject this view, I retain, so far 
as symmetrical transitive relations are concerned, what is really a some- 
what modified statement of the traditional doctrine. Such relations, to 
adopt more usual phraseology, are always constituted by possession of 
a common property. But a common property is not a very precise 
conception, and will not, in most of its ordinary significations, formally 
fulfil the function of analyzing the relations in question. A common 
quality of two terms is usually regarded as a predicate of those terms. 
But the whole doctrine of subject and predicate, as the only form of 
which propositions are capable, and the whole denial of the ultimate 
reality of relations, are rejected by the logic advocated in the present 
work. Abandoning the word predicate, we may say that the most 
general sense which can be given to a common property is this: A 
common property of two terms is any third term to which both have 
one and the same relation. In this general sense, the possession of 
a common property is symmetrical, but not necessarily transitive. In 
order that it may be transitive, the relation to the common property 
must be such that only one term at most can be the property of any 
given term*. Such is the relation of a quantity to its magnitude, or of 
an event to the time at which it occurs : given one term of the relation j 
namely the referent, the other is determinate, but given the other, the 
one is by no means determinate. Thus it is capable of demonstration 
that the possession of a common property of the type in question always 

* The proof of these assertions is mathematical^ and depends upon the Logic 
of Relations; it will be found in my article '^'^Sur la Logique des Relations/' 
B. d. M. VII, No. 2, § 1, Props. 6. 1, and 6. 2. 



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156-158] The meaning of Magnitude 167 

leads to a symmetrical transitive relation. What the principle of 
abstraction asserts is the converse, that such relations only spring from 
common properties of the above type *. It should be observed that the 
relation of the terms to what I have called their common property can 
never be that which is usually indicated by the relation of subject to 
predicate, or of the individual to its class. For no subject (in the 
received view) can have only one predicate, and no individual can belong 
to only one class. The relation of the terms to their common property 
is, in general, different in different cases. In the present case, the 
quantity is a complex of which the magnitude forms an element : the 
relation of the quantity to the magnitude is further defined by the 
fact that the magnitude has to belong to a certain class, namely that of 
magnitudes. It must then be taken as an axiom (as in the case of 
colours) that two magnitudes of the same kind cannot coexist in one 
spatio-temporal place, or subsist as relations between the same pair of 
terms; and this supplies the required uniqueness of the magnitude. It 
is such synthetic judgments of incompatibility that lead to negative 
judgments; but this is a purely logical topic, upon which it is not 
necessary to enlarge in this connection. 

158. We may now sum up the above discussion in a brief statement 
of results. There are a certain pair of indefinable relations, called 
greater and less ; these relations are asymmetrical and transitive, and 
are inconsistent the one with the other. Each is the converse of the 
other, in the sense that, whenever the one holds between A and B^ the 
other holds between B and J. The terms which are capable of these 
relations are magnitudes. Every magnitude has a certain peculiar 
relation to some concept, expressed by saying that it is a magnitude o/*that 
concept. Two magnitudes which have this relation to the same concept 
are said to be of the same kind ; to be of the same kind is the necessary 
and sufficient condition for the relations of greater and less. When a 
niagnitude can be particularized by temporal, spatial, or spatio-temporal 
position, or when, being a relation, it can be particularized by taking 
into a consideration a pai?- of terms between which it holds, then the 
magnitude so. particularized is called a quantity. Two magnitudes of 
the same kind can never be particularized by exactly the same specifi- 
cations. Two quantities which result from particularizing the same 
magnitude are said to be equal. 

Thus our indefinables are (1) greater and less, (^) every particular 
magnitude. Our indemonstrable propositions are : 

* The principle is proved by showing that, if R be a symmetrical transitive 
relation, and a a term of the field of B, a has, to the class of terms to which it has 
the relation B taken as a whole, a many-one relation which, relationally multiplied 
by its converse, is equal to R. Thus a magnitude may, so far as formal arguments 
are concerned, be identified with a class of equal quantities. 



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168 Quantity [CHAP. xix 

(1) Every magnitude has to some term the relation which makes 
it of a certain kind. 

(^) Any two magnitudes of the same kind are one greater and the 
other less. 

(3) Two magnitudes of the same kind, if capable of occupying space 
or time, cannot both have the same spatio-temporal position ; if relations, 
can never be both relations between the same pair of terms. 

(4) No magnitude is greater than itself. 

(5) If A is greater than B^ B is less than A^ and vice versa. 

(6) If -i is greater than B and B is greater than C, then A is 
greater than C*. 

Further axioms characterize various species of magnitudes, but the 
above seem alone necessary to magnitude in general. None of them 
depend in any way upon number or measurement ; hence we may be 
undismayed in the presence of magnitudes which cannot be divided or 
measured, of which, in the next chapter, we shall find an abundance of 
instances. 

Note to Chapter XIX. The work of Herr Meinong on Weber's Law, 
already alluded to, is one from , which I have learnt so much, and with 
which I so largely agree, that it seems desirable to justify myself on 
the points in which I depart from it. This work begins (§1) by a 
characterization of magnitude as that which is limited towards zero. 
Zero is understood as the negation of magnitude, and after a discussion, 
the following statement is adopted (p. 8) : 

" That is or has magnitude, which allows the interpolation of terms 
between itself and its contradictory opposite."' 

Whether this constitutes a definition, or a mere criterion, is left 
doubtful (i^.), but in either case, it appears to me to be undesirable as 
a fundamental characterization of magnitude. It derives support, as 
Herr Meinong points out (p. 6 n.\ from its similarity to Kant's 
"Anticipations of Perception t.'' But it is, if I am not mistaken, liable 
to several grave objections. In the first place, the whole theory of zero 
is most difficult, and seems subsequent, rather than prior, to the theory 
of other magnitudes. And to regard zero as the contradictory opposite 
of other magnitudes seems erroneous. The phrase should denote the 
class obtained by negation of the class "magnitudes of such and such 
a kind"; biit this obviously would not yield the zero of that kind of 
magnitude. Whatever interpretation we give to the phrase, it would 
seem to imply that we must regard zero as not a magnitude of the kind 
whose zero it is. But in that case it is not less than the magnitudes of 
the kind in question, and there seems no particular meaning in saying 

* It is not necessary in (5) and (6) to add '' A, B, G being magnitudes/' for the 
above relations of greater and less are what define magnitudes^ and the addition 
would therefore be tautological. 

t Reine Vernunft, ed. Hartenstein (1867), p. 1-58. 



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158] T'he meaning of Magnitude 169 

that a lesser magnitude is between zero and a greater magnitude. And 
in any case, the notion of between^ as we shall see in Part IV, demands 
asymmetrical relations among the terms concerned. These relations, it 
would seem, are, in the case of magnitude, none other than greater and 
less^ which are therefore prior to the betweenness of magnitudes, and 
more suitable to definition. I shall endeavour at a later stage to give 
what I conceive to be the true theory of zero ; and it will then appear 
how difficult this subject is. It can hardly be wise, therefore, to introduce 
zero in the first account of magnitude. Other objections might be urged, 
as, for instance, that it is doubtful whether all kinds of magnitude have 
a zero ; that in discrete kinds of magnitude, zero is unimportant ; and 
that among distances, where the zero is simply identity, there is hardly 
the same relation of zero to negation or non-existence as in the case of 
qualities such as pleasure. But the main reason must be the logical 
inversion involved in the introduction of betiveen before any asymmetrical 
relations have been specified from which it could arise. This subject 
will be resumed in Chapter xxii. 



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CHAPTER XX. 



THE RANGE OF QUANTITY. 

159. The questions to be discussed in the present chapter are these : 
What kinds of terms are there which, by their common relation to a 
number of magnitudes, constitute a class of quantities of one kind? 
Have all such terms anything else in common ? Is there any mark 
which will ensure that a term is thus related to a set of magnitudes ? 
What sorts of terms are capable of degree, or intensity, or greater and 
less ? 

The traditional view regards divisibility as a common mark of all 
terms having magnitude. We have already seen that there is no 
a priori ground for this view. We are now to examine the question 
inductively, to find as many undoubted instances of quantities as possible, 
and to inquire whether they all have divisibility or any other common 
mark. 

Any term of which a greater or less degree is possible contains under 
it a collection of magnitudes of one kind. Hence the comparative form 
in grammar is prima facie evidence of quantity. If this evidence were 
conclusive, we should have to admit that all, or almost all, qualities are 
susceptible of magnitude. The praises and reproaches addressed by 
poets to their mistresses would afford comparatives and superlatives 
of most known adjectives. But some circumspection is required in 
using evidence of this grammatical nature. There is always, I think, 
some quantitative comparison wherever a comparative or superlative 
occurs, but it is often not a comparison as regards the quality indicated 
by grammar. 

^^O ruddier than the cherry^ 
O sweeter than the berry_, 
O nymph more bright 
Than moonshine light/' 

are lines containing three comparatives. As regards sweetness and 
brightness, we have, I think, a genuine quantitative comparison ; but as 
regards ruddiness, this may be doubted. The comparative here — and 
generally where colours are concerned — indicates, I think, not more of a 



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159, 160] The range of Quantity l7l 

given colour, but more likeness to a standard colour. Various shades of 
colour are supposed to be arranged in a series, such that the difference 
of quality is greater or less according as the distance in the series is 
greater or less. One of these shades is the ideal " ruddiness,**'' and others 
are called more or less ruddy according as they are nearer to or further 
from this shade in the series. The same explanation applies, I think, 
to such terms as whiter^ blacker, redder. The true quantity involved 
seems to be, in all these cases, a relation, namely the relation of similarity. 
The difference between two shades of colour is certainly a difference of 
quality, not merely of magnitude ; and when we say that one thing is 
redder than another, we do not imply that the two are of the same shade. 
If there were no difference of shade, we should probably say one was 
brighter than the other, which is quite a different kind of comparison. 
But though the difference of two shades is a difference of quality, yet, as 
the possibility of serial arrangement shows, this difference of quality is 
itself susceptible of degrees. Each shade of colour seems to be simple 
and unanalyzable ; but neighbouring colours in the spectrum are certainly 
more similar than remote colours. It is this that gives continuity to 
colours. Between two shades of colour, A and J5, we should say, there 
is always a third colour C ; and this means that C resembles A or B 
more than B or A does. But for such relations of immediate resemblance, 
we should not be able to arrange colours in series. The resemblance 
must be immediate, since all shades of colour are unanalyzable, as appears 
from any attempt at description or definition*. Thus we have an 
indubitable case of relations which have magnitude. The difference or 
resemblance of two colours is a relation, and is a magnitude; for it is 
greater or less than other differences or resemblances. 

160. I have dwelt upon this case of colours, since it is one instance 
of a very important class. When any number of terms can be arranged 
in a series, it frequently happens that any two of them have a relation 
which may, in a generalized sense, be called a distance. This relation 
suffices to generate a serial arrangement, and is always necessarily a 
magnitude. In all such cases, if the terms of the series have names, and 
if these names have comparatives, the comparatives indicate, not more 
of the term in question, but more likeness to that term. Thus, if we 
suppose the time-series to be one in which there is distance, when an 
event is said to be more recent than another, what is meant is that its 
distance from the present was less than that of the other. Thus recentness 
is not itself a quality of the time or of the event. What are quantitatively 

"^ On the subject of the resemblances of colours^ see Meinong^ ^^'^ Abstrahiren und 
Vergleichen/' Zeitschrift f. Psych, u. Phys. d. Sinnesorgane, Vol. xxiv, p. 72 if. 
I am not sure that I agree with the whole of Meinong's argument, but his general 
conclusion, '^'^dass die UmfangscoUective des Aehnlichen AUgemeinheiten darstellen, 
an denen die Abstraction wenigstens unmittelbar keinen Antheii hat" (p. 78), 
appears to me to be a correct and important logical principle. 



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172 Quantity [char xx 

compared in such cases are relations, not qualities. The case of colours 
is convenient for illustration, because colours have names, and the 
difference of two colours is generally admitted to be qualitative. But 
the principle is of very wide application. The importance of this class 
of magnitudes, and the absolute necessity of clear notions as to their 
nature, will appear more and more as we proceed. The whole philosophy 
of space and time, and the doctrine of so-called extensive magnitudes, 
depend throughout upon a clear understanding of series and distance. 

Distance must be distinguished from mere difference or unlikeness. 
It holds only between terms in a series. It is intimately connected with 
order, and implies that the terms between which it holds have an ultimate 
and simple difference, not one capable of analysis into constituents. 
It implies also that there is a more or less continuous passage, through 
other terms belonging to the same series, from one of the distant terms 
to the other. Mere difference per se appears to be the bare minimum of 
a relation, being in fact a precondition of almost all relations. It is 
always absolute, and is incapable of degrees. Moreover it holds between 
any two terms whatever, and is hardly to be distinguished from the 
assertion that they are two. But distance holds only between the 
members of certain series, and its existence is then the source of the 
series. It is a specific relation, and it has senses we can distinguish 
the distance of A from B from that of B from A. This last mark 
alone suffices to distinguish distance from bare difference. 

It might perhaps be supposed that, in a series in which there is 
distance, although the distance AB must be greater than or less than AC^ 
yet the distance BD need not be either greater or less than AC. For 
example, there is obviously more difference between the pleasure 
derivable from £5 and that derivable from £100 than between that 
from £5 and that from £^Z0. But need there be either equality or 
inequality between the difference for £1 and £^0 and that for £5 and 
c£^100? This question must be answered affirmatively. For AC h 
greater or less than J5C, and BC is greater or less than BD ; hence A C, 
BC and also BC^ BD are magnitudes of the same kind. Hence A C, BD 
are magnitudes of the same kind, and if not identical, one must be the 
greater and the other the less. Hence, when there is distance in a series, 
any two distances are quantitatively comparable. 

It should be observed that all the magnitudes of one kind form 
a series, and that their distances, therefore, if they have distances, are 
again magnitudes. But it must not be supposed that these can, in 
general, be obtained by subtraction, or are of the same kind as the 
magnitudes whose differences they express. Subtraction depends, as a 
rule, upon divisibility, and is therefore in general inapplicable to 
indivisible quantities. The point is important, and will be treated 
in detail in the following chapter. 

Thus nearness and distance are relations which have magnitude. 



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160-162] The range of Quantity 173 

Are there any other relations having magnitude ? This may, I think, 
be doubted*. At least I am unaware of any other such relation, though 
I know no way of disproving their existence. 

161. There is a difficult class of terms, usually regarded as mag- 
nitudes, apparently implying relations, though certainly not always 
relational. These are differential coefficients, such as velocity and 
acceleration. They must be borne in mind in all attempts to generalize 
about magnitude, but owing to their complexity they require a special 
discussion. This will be given in Part V ; and we shall then find that 
differential coefficients are never magnitudes, but only real numbers, or 
segments in some series. 

162. All the magnitudes dealt with hitherto have been, strictly 
speaking, indivisible. Thus the question arises : Are there any divisible 
magnitudes ? Here I think a distinction must be made. A magnitude 
is essentially one, not many. Thus no magnitude is correctly expressed 
as a number of terms. But may not the quantity which has magnitude 
be a sum of parts, and the magnitude a magnitude of divisibility ? If so, 
every whole consisting of parts will be a single term possessed of the pro- 
perty of divisibility. The more parts it consists of, the greater is its 
divisibility. On this supposition, divisibility is a magnitude, of which we 
may have a greater or less degree ; and the degree of divisibility corresponds 
exactly, in finite wholes, to the number of parts. But though the whole 
which has divisibility is of course divisible, yet its divisibility, which alone 
is strictly a magnitude, is not properly speaking divisible. The divisibility 
does not itself consist of parts, but only of the property of having parts. 
It is necessary, in order to obtain divisibility, to take the whole strictly 
as 07ie^ and to regard divisibility as its adjective. Thus although, in 
this case, we have numerical measurement, and all the mathematical 
consequences of division, yet, philosophically speaking, our magnitude is 
still indivisible. 

There are difficulties, however, in the way of admitting divisibility as 
a kind of magnitude. It seems to be not a property of the whole, but 
merely a relation to the parts. It is difficult to decide this point, but a 
good deal may be said, I think, in support of divisibility as a simple 
quality. The whole has a certain relation, which for convenience we may 
call that of inclusion, to all its parts. This relation is the same whether 
there be many parts or few ; what distinguishes a whole of many parts is 
that it has many such relations of inclusion. But it seems reasonable to 
suppose that a whole of many parts differs from a whole of few parts in 
some intrinsic respect. In fact, wholes may be arranged in a series 
according as they have more or fewer parts, and the serial arrangement 
implies, as we have already seen, some series of properties differing more 
or less from each other, and agreeing when two wholes have the same 

* Cf. Meinoiig, Uebei* die Bedeutung des Weherschen Gesetzes, Hamburg and 
Leipzig, 1896, p. 23. 



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174 Quantity [ohap. xx 

finite number of parts, but distinct from number of parts in finite 
wholes. These properties can be none other than greater or less degrees 
of divisibility. Thus magnitude of divisibility would appear to be a 
simple property of a whole, distinct from the number of parts included 
in the whole, but correlated with it, provided this number be finite. If 
this view can be maintained, divisibility may be allowed to remain as a 
numerically measurable, but not divisible, class of magnitudes. In this 
class we should have to place lengths, areas and volumes, but not 
distances. At a later stage, however, we shall find that the divisibility 
of infinite wholes, in the sense in which this is not measured by cardinal 
numbers, must be derived through relations in a way analogous to that 
in which distance is derived, and must be really a property of relations*. 

Thus it would appear, in any case, that all magnitudes are in- 
divisible. This is one common mark which they all possess, and so far 
as I know, it is the only one to be added to those enumerated in 
Chapter xix. Concerning the range of quantity, there seems to be no 
further general proposition. Very many simple non-relational terms 
have magnitude, the principal exceptions being colours, points, instants 
and numbers. 

163. Finally, it is important to remember that, on the theory 
adopted in Chapter xix, a given magnitude of a given kind is a simple 
concept, having to the kind a relation analogous to that of inclusion in 
a class. When the kind is a kind of existents, such as pleasure, what 
actually exists is never the kind, but various particular magnitudes of 
the kind. Pleasure, abstractly taken, does not exist, but various amounts 
of it exist. This degree of abstraction is essential to the theory of 
quantity : there must be entities which differ from each other in nothing 
except magnitude. The grounds for the theory adopted may perhaps 
appear more clearly from a further examination of this case. 

Let us start with Bentham's famous proposition : " Quantity of 
pleasure being equal, pushpin is as good as poetry.*'*' Here the qualita- 
tive difference of the pleasures is the very point of the judgment ; but in 
order to be able to say that the quantities of pleasure are equal, we 
must be able to abstract the qualitative differences, and leave a certain 
magnitude of pleasure. If this abstraction is legitimate, the qualitative 
difference must be not truly a difference of quality, but only a difference 
of relation to other terms, as, in the present case, a difference in the 
causal relation. For it is not the whole pleasurable states that are 
compared, but only— as the form of the judgment aptly illustrates — 
their quality of pleasure. If we suppose the magnitude of pleasure to 
be not a separate entity, a difficulty will arise. For the mere element of 
pleasure must be identical in the two cases, whereas we require a possible 
difference of magnitude. Hence we can neither hold that only the 
whole concrete state exists, and any part of it is an abstraction, nor that 

* See Chap, xlvii. 



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162, 163] The range of Quantity 176 

what exists is abstract pleasure, not magnitude of pleasure. Nor can we 
say : We abstract, from the whole states, the two elements magnitude 
and pleasure. For then we should not get a quantitative comparison of 
the pleasures. The two states would agree in being pleasures, and in 
being magnitudes. But this would not give us a magnitude of pleasure ; 
and it would give a magnitude to the states as a whole, which is not 
admissible. Hence we cannot abstract magnitude in general from the 
states, since as wholes they have no magnitude. And we have seen that 
we must not abstract bare pleasure, if we are to have any possibility of 
different magnitudes. Thus what we have to abstract is a magnitude of 
pleasure las a whole. This must not be analyzed into magnitude and 
pleasure, but must be abstracted as a whole. And the magnitude of 
pleasure must exist as a part of the whole pleasurable states, for it is 
only where there is no difference save at most one of magnitude that 
quantitative comparison is possible. Thus the discussion of this parti- 
cular case fully confirms the theory that every magnitude is unanalyzable, 
and has only the relation analogous to inclusion in a class to that 
abstract quality or relation of which it is a magnitude. 

Having seen that all magnitudes are indivisible, we have next to 
consider the extent to which numbers can be used to express magnitudes, 
and the nature and limits of measurement. 



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CHAPTER XXI. 

NUMBERS AS EXPRESSING MAGNITUDES: 
MEASUREMENT. 

164. It is one of the assumptions of educated common-sense that 
two magnitudes of the same kind must be numerically comparable. 
People are apt to say that they are thirty per cent, healthier or happier 
than they were, without any suspicion that such phrases are destitute of 
meaning. The purpose of the present chapter is to explain what is 
meant by measurement, what are the classes of magnitudes to which it 
applies, and how it is applied to those classes. 

Measurement of magnitudes is, in its most general sense, any method 
by which a unique and reciprocal correspondence is established between 
all or some of the magnitudes of a kind and all or some of the numbers, 
integral, rational, or real, as the case may be. (It might be thought 
that complex numbers ought to be included ; but what can only be 
measured by complex numbers is in fact always an aggregate of magni- 
tudes of different kinds, not a single magnitude.) In this general sense, 
measurement demands some one-one relation between the numbers and 
magnitudes in question — a relation which may be direct or indirect, 
important or trivial, according to circumstances. Measurement in this 
sense can be applied to very many classes of magnitudes ; to two great 
classes, distances and divisibilities, it applies, as we shall see, in a more 
important and intimate sense. 

Concerning measurement in the most general sense, there is very 
little to be said. ' Since the numbers form a seines, and since every kind 
of magnitude also forms a series, it will be desirable that the order of 
the magnitudes measured should correspond to that of the numbers, i.e. 
that all relations of between should be the same for magnitudes and their 
measures. Wherever there is a zero, it is well that this should be 
measured by the number zero. , These and other conditions, which a 
measure should fulfil if possible, may be laid down; but they are of 
practical rather than theoretical importance. 

165. There are two general metaphysical opinions, either of which, 
if accepted, shows that all magnitudes are theoretically capable of 
measurement in the above sense. The first of these is the theory that 



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164-166] Measurement 177 

all events either are, or are correlated with, events in the dynamical 
causal series. In regard to the so-called secondary qualities, this view 
has been so far acted upon by physical science that it has provided most 
of the so-called intensive quantities that appear in space with spatial, 
and thence numerical, measures. And with regard to mental quantities 
the theory in question is that of psychophysical parallelism. Here the 
motion which is Correlated with any pjsychical quantity always theoreti- 
cally affords a means of measuring that quantity. The other metaphysical 
opinion, which leads to universal measurability, is one suggested by 
Kant's "Anticipations of Perception*," namely that, among intensive 
magnitudes, an increase is always accompanied by an increase of reality. 
Reality, in this connection, seems synonymous with existence ; hence 
the doctrine may be stated thus : Existence is a kind of intensive 
magnitude, of which, where a greater magnitude exists, there is always 
more than where a less magnitude exists. (That this is exactly Kant's 
doctrine seems improbable; but it is at least a tenable view.) In this 
case, since two instances of the same magnitude {i.e. two equal quantities) 
must have more existence than one, it follows that, if a single magnitude 
of the same kind can be found having the same amount of existence as 
the two equal quantities together, then that magnitude may be called 
double that of each of the equal quantities. In this way all intensive 
magnitudes become theoretically capable of measurement. That this 
method has any practical importance it would be absurd to maintain ; 
but it may contribute to the appearance of meaning belonging to txmce 
as happy. It gives a sense, for example, in which we may say that a 
child derives as much pleasure from one chocolate as from two acid 
drops ; and on the basis of such judgments the hedonistic Calculus 
could theoretically be built. 

There is one other general observation of some importance. If it be 
maintained that all series of magnitudes are either continuous in Cantor's 
sense, or are similar to series which can be chosen out of continuous 
series, then it is theoretically possible to correlate any kind of magnitudes 
with all or some of the real numbers, so that the zeros correspond, and 
the greater magnitudes correspond to the greater numbers. But if any 
series of magnitudes, without being continuous, contains continuous 
series, then such a series of magnitudes will be strictly and theoretically 
incapable of measurement by the real numbers f. 

166. Leaving now these somewhat vague generalities, let us examine 
the more usual and concrete sense of measurement. AVhat we require is 
some sense in which we may say that one magnitude is double of another. 



■^ Reine Vernunft, ed. Hart. (1867):, P- 1^0. The wording of the first edition 
illustrates better than that of the second the doctrine to which I allude. See e.g. 
Erdmann's edition^ p. 161. 

+ See Part V^ Chap, xxxm if. 

R. 12 



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178 Quantity [chap, xxi 

In the above instances this sense was derived by correlation with spatio- 
temporal magnitudes, or with existence. This presupposed that in these 
cases a meaning had been found for the phrase. Hence measurement 
demands that, in some cases, there should be an intrinsic meaning to the 
proposition ''this magnitude is double of that.'' (In what sense the 
meaning is intrinsic will appear as we proceed.) Now so long as 
quantities are regarded as inherently divisible, there is a perfectly 
obvious meaning to such a proposition : a magnitude A is double of B 
when it is the magnitude of two quantities together, each of these 
having the magnitude B. (It should be observed that to divide a 
magnitude into two equal parts must always be impossible, since there 
are no such things as equal magnitudes.) Such an interpretation will 
still apply to magnitudes of divisibility ; but since we have admitted 
other magnitudes, a different interpretation (if any) must be found for 
these. Let us first examine the case of divisibilities, and then proceed 
to the other cases where measurement is intrinsically possible. 

167. The divisibility of a finite whole is immediately and inherently 
correlated with the number of simple parts in the whole. In this case, 
although the magnitudes are even now incapable of addition of the sort 
required, the quantities can be added in the manner explained in Part II. 
The addition of two magnitudes of divisibility yields merely two magni- 
tudes, not a new magnitude. But the addition of two quantities of 
divisibility, i.e. two wholes, does yield a new single whole, provided the 
addition is of the kind which results from logical addition by regarding 
classes as the wholes formed by their terms. Thus there is a good 
meaning in saying that one magnitude of divisibility is double of 
another, when it applies to a whole containing twice as many parts. 
But in the case of infinite wholes, the matter is by no means so simple. 
Here the number of simple parts (in the only senses of infinite number 
hitherto discovered) may be equal without equality in the magnitude of 
divisibility. We require here a method which does not go back to 
simple parts. In actual space, we have immediate judgments of equality 
as regards two infinite wholes. When we have such judgments, we can 
regard the sum of n equal wholes as n times each of them ; for addition 
of wholes does not demand their finitude. In this way numerical com- 
parison of some pairs of wholes becomes possible. By the usual well- 
known methods, by continual subdivision and the method of limits, this 
is extended to all pairs of wholes which are such that immediate com- 
parisons are possible. Without these immediate comparisons, which 
are necessary both logically and psychologically*, nothing can be 
accomplished : we are always reduced in the last resort to the immediate 
judgment that our foot-rule has not greatly changed its size during 
measurement, and this judgment is prior to the results of physical 

* Cf, Meinong;, op. cit., pp. 63-4. 



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1 6 6, 1 6 7] Measureme7it 179 

science as to the extent to which bodies do actually change their sizes. 
But where immediate comparison is psychologically impossible, we may 
theoretically substitute a logical variety of measurement, which, however, 
gives a property not of the divisible whole, but of some relation or class 
of relations more or less analogous to those that hold between points in 
space. 

That divisibility, in the sense required for areas and volumes, is not 
a property of a whole, results from the fact (which will be established in 
Part VI) that between the points of a space there are alw^ays relations 
which generate a diiferent space. Thus two sets of points which, with 
regard to one set of relations, form equal areas, form unequal areas with 
respect to another set, or even form one an area and the other a line or 
a volume. If divisibility in the relevant sense were an intrinsic property 
of wholes, this would be impossible. But this subject cannot be fully 
discussed until we come to Metrical Geometry. 

Where our magnitudes are divisibilities, not only do numbers measure 
them, but the difference of two measuring numbers^ with certain limita- 
tions, measures the magnitude of the difference (in the sense of dis- 
similarity) between the divisibilities. If one of the magnitudes be 
fixed, its difference from the other increases as the difference of the 
measuring numbers increases ; for this difference depends upon the 
difference in the number of parts. But I do not think it can be shown 
generally that, if J, jB, C, Z> be the numbers measuring four magnitudes, 
Qjudi A —B=C — D^ then the differences of the magnitvides are equal. 
It would seem, for instance, that the difference between one inch and 
two inches is greater than that between 1001 inches and 1002 inches. 
This remark has no importance in the present case, since differences of 
divisibility are never required ; but in the case of distances it has a 
curious connection with non-Euclidean Geometry. But it is theoretically 
important to observe that, if divisibility be indeed a magnitude — as the 
equality of areas and volumes seems to require — then there is strictly no 
ground for saying that the divisibility of a sum of two units is twice as 
great as that of one unit. Indeed this proposition cannot be strictly 
taken, for no magnitude is a sum of parts, and no magnitude therefore is 
double of another. We can only mean that the sum of two units con- 
tains twice as many parts, which is an arithmetical, not a quantitative, 
judgment, and is adequate only in the case where the number of parts is 
finite, since in other cases the double of a number is in general equal to 
it. Thus even the measurement of divisibility by numbers contains 
an element of convention ; and this elementj we shall find, is still more 
prominent in the case of distances. 

168. In the above case we still had addition in one of its two 
fundamental senses, i,e. the combination of wholes to form a new whole. 
But in other cases of magnitude we do not have any such addition. 
The sum of two pleasures is not a new pleasure, but is merely two 

12^2 



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180 Quantity [chap, xxi 

pleasures. The sum of two distances is also not properly one distance. 
But in this case we have an extension of the idea of addition. Some 
such extension must always be possible where measurement is to be 
effected in the more natural and restricted sense which we are now 
discussing. I shall first explain this generalized addition in abstract 
terms, and then illustrate its application to distances. 

It sometimes happens that two quantities, which are not capable of 
addition proper, have a relation, which has itself a one-one relation to 
a quantity of the same kind as those between which it holds. Supposing 
a^ 6, c to be such quantities, we have, in the case supposed, some pro- 
position aBc^ where jB is a relation which uniquely determines and is 
uniquely determined by some quantity h of the same kind as that to 
which a and c belong. Thus for example two ratios have a relation, 
which we may call their difference, which is itself wholly determined by 
another ratio, namely the difference, in the arithmetical sense, of the 
two given ratios. If a, yS, 7 be terms in a series in which there is 
distance, the distances ajB^ a^y have a relation which is measured by 
(though not identical with) the distance Z^^. In all such cases, by an 
extension of addition, we may put a + b = c in place of aBc. Wherever 
a set of quantities have relations of this kind, if further aBc implies bAc, 
so that a -\-h = h-\-a^ we shall be able to proceed as if we had ordinary 
addition, and shall be able in consequence to introduce numerical 
measurement. 

The conception of distance will be discussed fully in Part IV, in 
connection with order : for the present I am concerned only to show 
how distances come to be measurable. The word will be used to cover 
a far more general conception than that of distance in space. I shall 
mean by a kind of distance a set of quantitative asymmetrical relations of 
which one and only one holds between any pair of terms of a given 
class ; which are such that, if there is a relation of the kind between a 
and &, and also between h and c^ then there is one of the kind between 
a and c^ the relation between a and c being the relative product of 
those between a and &, & and c; this product is to be commutative, 
i. e, independent of the order of its factors ; and finally, if the distance 
ab be greater than the distance ac^ then, d being any other member of 
the class, dh is greater than dc. Although distances are thus relations, 
and therefore indivisible and incapable of addition proper, there is a 
simple and natural convention by which such distances become numeri- 
cally measurable. 

The convention is this. Let it be agreed that, when the distances 
^0^1? <^i^2 ••• CLn-idn ^^c all equal and in the same sense, then a^an is 
said to be n times each of the distances <^o^i? etc., i. e, is to be measured 
by a number n times as great. This has generally been regarded as not 
a convention, but an obvious truth ; owing, however, to the fact that 
distances are indivisible, no distance is really a sum of other distances. 



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167-170] Measurement 181 

and numerical measurement must be in part conventional. With this 
convention, the numbers corresponding to distances, where there are 
such numbers, become definite, except as to a common factor dependent 
upon the choice of a unit. Numbers are also assigned by this method 
to the members of the class between which the distances hold; these 
numbers have, in addition to the arbitrary factor, an arbitrary additive 
constant, depending upon the choice of origin. This method, which is 
capable of still further generalization, will be more fully explained in 
Part iV. In order to show that all the distances of our kind, and all 
the terms of our set, can have numbers assigned to them, we require two 
further axioms, the axiom of Archimedes, and what may be called the 
axiom of linearity*. 

169. The importance of the numerical measurement of distance, at 
least as applied to space and time, depends partly upon a further fact, 
by which it is brought into relation with the numerical measurement of 
divisibility. In all series there are terms intermediate between any two 
whose distance is not the minimum. These terms are determinate when 
the two distant terms are specified. The intermediate terms may be 
called the stretch from a^ to «^^*t*. The whole composed of these terms 
is a quantity, and has a divisibility measured by the number of terms, 
provided their number is finite. If the series is such that the distances 
of consecutive terms are all equal, then, if there are n—l terms between 
a^ and ^^, the measure of the distance is proportional to n. Thus, if we 
include in the stretch one of the end terms, but not the other, the 
measures of the stretch and the distance are proportional, and equal 
stretches correspond to equal distances. Thus the number of terms in 
the stretch measures both the distance of the end terras and the amount 
of divisibility of the whole stretch. When the stretch contains an 
infinite number of terms, we estimate equal stretches as explained above. 
It then becomes an axiom, which may or may not hold in a given case, 
that equal stretches correspond to equal distances. In this case, co- 
ordinates 'measure two entirely distinct magnitudes, which, owing to 
their common measure, are perpetually confounded. 

170. The above analysis explains a curious problem which must 
have troubled most people who have endeavoured to philosophize about 
Geometry. Starting frora one-dimensional magnitudes connected with 
the straight line, most theories may be divided into two classes, those 
appropriate to areas and volumes, and those appropriate to angles 

"^ See Part IV^ Chap. xxxi. This axiom asserts that a magnitude can be divided 
into n equal parts^ and forms part of Du Bois Reymond's definition of linear magni- 
tude. See his AUgemeine Fanctionentheorie (Tubingen^ 1882)^ Chap, i, § 16 ; also 
Bettazzi_, Teoria delle Grande zase (Pisa^ 1890)_, p. 44. The axiom of Archimedes 
asserts that_, given any two magnitudes of a kind_, some finite multiple of the lesser 
exceeds the greater. 

t Called Strecke by Meinong^ op, cit., e.g, p. 22. 



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182 Quantity [chap, xxi 

between lines or planes. Areas and volumes are radically different 
from angles, and are generally neglected in philosophies which hold 
to relational views of space or start from projective Geometry. The 
reason of this is plain enough. On the straight line, if, as is usually 
supposed, there is such a relation as distance, we have two philosophi- 
cally distinct but practically conjoined magnitudes, namely the distance, 
and the divisibility of the stretch. The former is similar to angles ; the 
latter, to areas and volumes. Angles may also be regarded as distances 
between terms in a series j namely between lines through a point or 
planes through a line., Areas and volumes, on the contrary, are sums, 
or magnitudes of divisibility. Owing to the confusion of the two kinds 
of magnitude connected with the line, either angles, or else areas and 
volumes, are usually incompatible with the philosophy invented to 
suit the line. By the above analysis, this incompatibility is at once 
explained and overcome*. 

171. We thus see how two great classes of magnitudes — divisibilities 
and distances — are rendered amenable to measure. These two classes 
practically cover what are usually called extensive magnitudes, and it 
will be convenient to continue to allow the name to them. I shall 
extend this name to cover all distances and divisibilities, whether they 
have any relation to space and time or not. But the word extensive 
must not be supposed to indicate, as it usually does, that the magnitudes 
so designated are divisible. We have already seen that no magnitude is 
divisible. Quantities are only divisible into other quantities in the one 
case of wholes which, are quantities of divisibility. Quantities which are 
distances, though I shall call them extensive, are not divisible into 
smaller distances; but they allow the important kind of addition ex- 
plained above, which I shall call in, future relational eiAdiition^;. 

AH other magnitudes and quantities may be properly called intensive. 
Concerning these, unless by some causal relation, or by means of some 
more or less roundabout relation such as those explained at the beginning 
of the present chapter, numerical measurement is impossible. Those 
mathematicians who are accustomed to an exclusive emphasis on numbers, 
will think that not much can be said with definiteness concerning magni- 
tudes incapable of measurement. This, however, is by no means the 
case. The immediate judgments of equality, upon which (as we saw) 
all measurements depend, are still possible where measurement fails, as 
are also the immediate judgments of greater and less. Doubt only 
arises where the difference is small ; and all that measurement does, 

* In Part VI^ we shall find reason to deny distance in most spaces. But there 
is still a distinction between stretches_, consisting of the terms of some series_, and 
such quantities as areas and volumes^ where the terms do not^ in any simple sense^ 
form a one-dimensional series, 

t Not to be confounded with the relative addition of the Algebra of Relatives. 
It is connected rather with relative multiplication. 



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170, 171] Measurement 183 

in this respect, is to make the margin of doubt smaller — an achievement 
which is purely psychological, and of no philosophical importance. 
Quantities not susceptible of numerical measurement can thus be ar- 
ranged in a scale of greater and smaller magnitudes, and this is the 
only strictly quantitative achievement of even numerical measurement. 
We can know that one magnitude is greater than another, and that 
a third is intermediate between them ; also, since the differences of 
magnitudes are always magnitudes, there is always (theoretically, at 
least) an answer to the question whether the difference of one pair 
of magnitudes is greater than, less than, or the same as the difference of 
another pair of the same kind. And such propositions, though to the 
mathematician they may appear approximate, are just as precise and 
definite as the propositions of Arithmetic. Without numerical measure- 
ment, therefore, the quantitative relations of magnitudes have all the 
definiteness of which they are capable — nothing is added, from the 
theoretical standpoint, by the assignment of correlated numbers. The 
whole subject of the measurement of quantities is, in fact, one of more 
practical than theoretical importance. What is theoretically important 
in it is merged in the wider question of the correlation of series, which 
will occupy us much hereafter. The chief reason why I have treated 
the subject thus at length is derived from its traditional importance, but 
for which it might have been far more summarily treated. 



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CHAPTER XXII. 

ZERO. 

172. The present chapter is concerned, not with any form of the 
numerical zero, nor yet with the infinitesimal, but with the pure zero 
of magnitude. This is the zero which Kant has in mind, in his refuta- 
tion of Mendelssohn's proof of the immortality of the soul *. Kant 
points out that an intensive magnitude, while remaining of the same 
kind, can become zero ; and that, though zero is a definite magnitude, 
no quantity whose magnitude is zero can exist. This kind of zero, we 
shall find, is a fundamental quantitative notion, and is one of the points 
in which the theory of quantity presents features peculiar to itself. The 
quantitative zero has a certain connection both with the number and 
with the null-class in Logic, but it is not (I think) definable in terms of 
either. What is less universally realized is its complete independence 
of the infinitesimal. The latter notion will not be discussed until the 
following chapter. 

The meaning of zero, in any kind of quantity, is a question of much 
difficulty, upon w^hich the greatest care must be bestowed, if contra- 
dictions are to be avoided. Zero seems to be definable by some general 
characteristic, without reference to any special peculiarity of the kind of 
quantity to which it belongs. To find such a definition, however, is far 
from easy. Zei^o seems to be a radically distinct conception according as 
the magnitudes concerned are discrete or continuous. To prove that 
this is not the case, let us examine various suggested definitions. 

173. (1) Herr Meinong {op, cit.^ p. 8) regards zero as the con- 
tradictory opposite of each magnitude of its kind. The phrase 
" contradictory opposite " is one which is not free from ambiguity. 
The opposite of a class, in symbolic logic, is the class containing all 
individuals not belonging to the first class ; and hence the opposite 
of an individual should be all other individuals. But this meaning is 
evidently inappropriate : zero is not everything except one magnitude 
of its kind, nor yet everything except the class of magnitudes of its 
kind. It can hardly be regarded as true to say that a pain is a zero 

■^ K7'itik der Reinen Vernunfty ed. Hartenstein^ p. 281 fF. 



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172-174] Zero 185 

pleasure. On the other hand, a zero pleasure is said to be no 'pleasure^ 
and this ds evidently what Herr Meinong means. But although we 
shall find this view to be correct, the meaning of the phrase is very 
difficult to seize. It does not mean something other than pleasure, 
as when our friends assure us that it is no pleasure to tell us our faults. 
It seems to mean what is neither pleasure, nor yet anything else. But 
this would be merely a cumbrous way of saying nothings and the 
reference to pleasure might be wholly dropped. This gives a zero 
which is the same for all kinds of magnitude, and if this be the true 
meaning of zero, then zero is not one among the magnitudes of a kind, 
nor yet a term in the series formed by magnitudes of a kind. For 
tholigh it is often true that there is nothing smaller than all the 
magnitudes of a kind, yet it is always false that nothing itself is 
smaller than all of them. This zero, therefore, has no special reference 
to any particular kind of magnitude, and is incapable of fulfilling the 
functions which Herr Meinong demands of it *. The phrase, however, 
as we shall see, is capable of an interpretation which avoids this difficulty. 
But let us first examine some other suggested meanings of the word. 

174. (^) Zero may be defined as the least magnitude of its kind. 
Where a kind of magnitude is discrete, and generally when it has what 
Professor Bettazzi calls a limiting magnitude of the kindf, such a 
definition is insufficient. For in such a case, the limiting magnitude 
seems to be really the least of its kind. And in any case, the definition 
gives rather a characteristic than a true definition, which must be sought 
in some more purely logical notion, for zero cannot fail to be in some 
sense a denial of all other magnitudes of the kind. The phrase that 
zero is the smallest of magnitudes is like the phrase which De Morgan 
commends for its rhetoric: "Achilles was the strongest of all his enemies." 
Thus it would be obviously false to say that is the least of the positive 
integers, or that the interval between A and A is the least interval 
between any two letters of the alphabet. On the other hand, where a 
kind of magnitude is continuous, and has no limiting magnitude, although 
we have apparently a gradual and unlimited approach to zero, yet now a 
new objection arises. Magnitudes of this kind are essentially such as 
have no minimum. Hence we cannot without express contradiction take 
zero as their minimum. We may, however, avoid this contradiction by 
saying that there is always a magnitude less than any other, but not 
zero, unless that other be zero. This emendation avoids any formal 
contradiction, and is only inadequate because it gives rather a mark of 
zero than its true meaning. Whatever else is a magnitude of the kind 
in question might have been diminished ; and we wish to know what it 
is that makes zero obviously incapable of any further diminution. This 
the suggested definition does not tell us, and therefore, though it gives a 

■^ See note to Chap, xix^ supi'a. 

t Teoria delle GrandezzSy Pisa^ 1890, p. 24. 



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186 Quantity [chap, xxii 

characteristic which often belongs to no other magnitude of the kind, it 
cannot be considered philosophically sufficient. Moreover, where there 
are negative magnitudes, it precludes us from regarding these as less 
than zero. 

175. (3) Where our magnitudes are differences or distances, zero 
has, at first sight, an obvious meaning, namely identity. But here, again, 
the zero so defined seems to have no relation to one kind of distances 
rather than another : a zero distance in time would seem to be the same 
as a zero distance in space. This can, however, be avoided, by substituting, 
for identity simply, identity with some member of the class of terms 
between >vhich the distances in question hold. By this device, the zero 
of any class of relations which are magnitudes is made perfectly definite 
and free from contradiction ; moreover we have both zero quantities and 
zero magnitudes, for \i A and ^ be terms of the class which has distances, 
identity with A and identity with 5 are distinct zero quantities*. This 
case, therefore, is thoroughly clear. And yet the definition must be 
rejected : for it is plain that zero has some general logical meaning, if 
only this could be clearly stated, which is the same for all classes of 
quantities; and that a zero distance is not actually the same concept as 
identity. 

176. (4) In any class of magnitudes which is continuous, in the 
sense of having a term between any two, and which also has no limiting 
magnitude, we can introduce zero in the manner in which real numbers are 
obtained from rationals. Any collection of magnitudes defines a class of 
magnitudes less than all of them. This class of magnitudes can be made as 
small as we please, and can actually be made to be the null-class, i.e. to 
contain no members at all. (This is effected, for instance, if our collection 
consists of all magnitudes of the kind.) The classes so defined form a 
series, closely related to the series of original magnitudes, and in this 
new series the null-class is definitely the first term. Thus taking the 
classes as quantities, the null-class is a zero quantity. There is no class 
containing a finite number of members, so that there is not, as in 
Arithmetic, a discrete approach to the null-class; on the contrary, the 
approach is (in several senses of the word) continuous. This method of 
defining zero, which is identical with that by which the real number zero 
is introduced, is important, and will be discussed in Part V. But for the 
present we may obsei've that it again makes zero the same for all kinds 
of magnitude, and makes it not one among the magnitudes whose zero 
it is. 

177. (5) We are compelled, in this question, to face the problem 
as to the nature of negation. " No pleasure ''' is obviously a different 
concept from "no pain,^' even when these terms are taken strictly as 
mere denials of pleasure and pain respectively. It would seem that "no 

■^ On this pointy however^ see § 6o above. 



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174-178] Zero 187 

pleasure ^^ has the same relation to pleasure as the various magnitudes of 
pleasure have, though it has also, of course, the special relation of 
negation. If this be allowed, we see that, if a kind of magnitudes be 
defined by that of which they are magnitudes, then no 'pleasure is one 
among the various magnitudes of pleasure. If, then, we are to hold to 
our axiom, that all pairs of magnitudes of one kind have relations of 
inequality, we shall be compelled to admit that zero is less than all other 
magnitudes of its kind. It seems, indeed, to be rendered evident that 
this must be admitted, by the fact that zero is obviously not greater 
than all other magnitudes of its kind. This shows that zero has a 
connection with less which it does not have with greater. And if we 
adopt this theory, we shall no longer accept the clear and simple account 
of zero distances given above, but we shall hold that a zero distance is 
strictly and merely no distance^ and is only correlated with identity. 

Thus it would seem that Herr Meinong's theory, with which we 
began, is substantially correct; it requires emendation, on the above 
view, only in this, that a zero magnitude is the denial of the defining 
concept of a kind of magnitudes, not the denial of any one particular 
magnitude, or of all of them. We shall have to hold that any concept 
which defines a kind of magnitudes defines also, by its negation, a 
particular magnitude of the kind, which is called the zero of that kind, 
and is less than all other members of the kind. And we now reap the 
benefit of the absolute distinction which we made between the defining 
concept of a kind of magnitude, and the various magnitudes of the kind. 
The relation which we allowed between a particular magnitude and that 
of which it is a magnitude was not identified with the class-relation, but 
was held to be sui generis ; there is thus no contradiction, as there 
would be in most, theories, in supposing this relation to hold between no 
pleasure said pleasure^ or between no distance waA distance. 

178. But finally, it must be observed that no pleasure^ the zero 
magnitude, is not obtained by the logical denial of pleasure, and is not' 
the same as the logical notion of not pleasure. On the contrary, no 
pleasure \^ essentially a quantitative concept, having a curious and 
intimate relation to logical denial, just as has a very intimate relation 
to the null-class; The relation is this, that there is no quantity whose 
magnitude is zero, so that the class of zero quantities is the null-class*. 
The zero of any kind of magnitude is incapable of that relation to 
existence or to particulars, of which the other magnitudes are capable. 
But this is a synthetic proposition, to be accepted only on account of its 
self-evidence. The zero magnitude of any kind, like the other magnitudes, 
is properly speaking indefinable, but is capable of specification by means 
of its peculiar relation to the logical zero. 

* This must be applied in correction of what was formerly said about zero 
distances. 



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CHAPTER XXIII. 



INFINITY, THE INFINITESIMAL, AND CONTINUITY. 

179. Almost all mathematical ideas present one great difficulty : 
the difficulty of infinity. This is usually regarded by philosophers as 
an antinomy, and as showing that the propositions of mathematics are 
not metaphysically true. From this received opinion I am compelled to 
dissent. Although all apparent antinomies, except such as are quite 
easily disposed of, and such as belong to the fundamentals of logic, are, 
in my opinion, reducible to the one difficulty of infinite number, yet this 
difficulty itself appears to be soluble by a correct philosophy of any^ and 
to have been generated very largely by confusions due to the ambiguity 
in the meaning of finite integers. The problem in general will be 
discussed in Part V ; the purpose of the present chapter is merely to 
show that quantity, which has been regarded as the true home of infinity, 
the infinitesimal, and continuity, must give place, in this respect, to 
order; while the statement of the difficulties which arise in regard to 
quantity can be made in a form which is at once ordinal and arithmetical, 
but involves no reference to the special peculiarities of quantity. 

180. The three problems of infinity, the infinitesimal, and con- 
tinuity, as they occur in connection with quantity, are closely related. 
None of them can be fully discussed at this stage, since all depend 
essentially upon order, while the infinitesimal depends also upon number. 
The question of infinite quantity, though traditionally considered more 
formidable than that of zero, is in reality far less so, and might be 
briefly disposed of, but for the great devotion commonly shown by 
philosophers to a proposition which I shall call the axiom of finitude. 
Of some kinds of magnitude (for example ratios, or distances in space 
and time), it appears to be true that there is a magnitude greater than 
any given magnitude. That is, any magnitude being mentioned, another 
can be found which is greater than it. The deduction of infinity from 
this fact is, when correctly performed, a mere fiction to facilitate com- 
pression in the statement of results obtained by the method of limits. 



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179-181] Infinity, the Infinitesimal, and Continuity 189 

Any class it of magnitudes of our kind being defined, three cases may 
arise : (1) There may be a class of terms greater than any of our class ii, 
and this new class of terms may have a smallest member ; (2) there may 
be such a class, but it may have no smallest member ; (3) there may be 
no magnitudes which are greater than any term of our class u. Suppos- 
ing our kind of magnitudes to be one in which there is no greatest 
magnitude, case (S) will always arise where the class u contains a finite 
number of terms. On the other hand, if our series be what is called 
condensed in itself, case (2) will never arise when u is an infinite class, 
and has no greatest term ; and if our series is not condensed in itself, 
but does have a term between any two, another which has this property 
can always be obtained from it*. Thus all infinite series which have 
no greatest term will have limits, except in case (3). To avoid cir- 
cumlocution, case (8) is defined as that in which the limit is infinite. 
But this is a mere device, and it is generally admitted by mathema- 
ticians to be such. Apart from special circumstances, there is no 
reason, merely because a kind of magnitudes has no maximum, to 
admit that there is an infinite magnitude of the kind, or that there 
are many such. When magnitudes of a kind having no maximum 
are capable of numerical measurement, they very often obey the axiom 
of Archimedes, in virtue of which the ratio of any two magnitudes of 
the kind is finite. Thus, so far, there might appear to be no problem 
connected with infinity. 

But at this point the philosopher is apt to step in, and to declare 
that, by all true philosophic principles, every well-defined series of terms 
must have a last term. If he insists upon creating this last term, and 
calling it infinity, he easily deduces intolerable contradictions, from which 
he infers the inadequacy of mathematics to obtain absolute truth. For 
my part, however, I see no reason for the philosopher's axiom. To show, 
if possible, that it is not a necessary philosophic principle, let us under- 
take its analysis, and see what it really involves. 

The problem of infinity, as it has now emerged, is not properly a 
quantitative problem, but rather one concerning order. It is only 
because our magnitudes form a series having no last term that the 
problem arises : the fact that the series is composed of magnitudes is 
wholly irrelevant. With this remark I might leave the subject to a 
later stage. But it will be worth while now to elicit, if not to examine, 
the philosopher's axiom of finitude. 

181. It will be well, in the first place, to show how the problem 
concerning infinity is the same as that concerning continuity and the 
infinitesimal. For this purpose, we shall find it convenient to ignore the 
absolute zero, and to mean, when we speak of any kind pf magnitudes, 
all the magnitudes of the kind except zero. This is a mere change of 

* This will be further explained in Part V_, Chap, xxxvi. 



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190 Quantity [chap, xxiii 

diction, without which intolerable repetitions would be necessary. Now 
there certainly are some kinds of magnitude where the three following 
axioms hold : 

(1) HA and B be any two magnitudes of the kind, and J is 
greater than B^ there is always a third magnitude C such that A 
is greater than C and C greater than B. (This I shall call, for the 
present, the axiom of continuity.) 

(2) There is always a magnitude less than any given magnitude B. 

(3) There is always a magnitude greater than any given magni- 
tude J. 

From these it follows : — 

(1) That no two magnitudes of the kind are consecutive. 

(S) That there is no least magnitude. 

(3) That there is no greatest magnitude. 

The above propositions are certainly true of some kinds of magni- 
tude ; whether they are true of all kinds remains to be examined. The 
following three propositions, which directly contradict the previous three, 
must be always true, if the philosopher's axiom of finitude is to be 
accepted : 

{a) There are consecutive magnitudes, i.e. magnitudes such that 
no other magnitude of the same kind is greater than the less and less 
than the greater of the two given magnitudes. 

{h) There is a magnitude smaller than any other of the same kind. 

(c) There is a magnitude greater than any other of the same 
kind*. 

As these three propositions directly contradict the previous three, it 
would seem that both sets cannot be true. We have to examine the 
grounds for both, and let one set of alternatives fall. 

182. Let us begin with the propositions («), (5), (c), and examine 
the nature of their grounds. 

{a) A definite magnitude A being given, all the magnitudes greater 
than A form a series, whose differences from A are magnitudes of a new 
kind. If there be a magnitude B consecutive to ^, its difference from A 
will be the least magnitude of its kind, provided equal stretches cor- 
respond to equal distances in the series. And conversely, if there be 
a smallest difference between two magnitudes, A^ jB, then these two 
magnitudes must always be consecutive ; for if not, any intermediate 

■^ Those Hegelians who search for a chance of an antinomy may proceed to 
the definition of zero and infinity by means of the above propositions. When (2) 
and {h) both hold^ they may say^ the magnitude satisfying {h) is called zero ; when 
(3) and (c) both hold^ the magnitude satisfying (c) is called infinity. We have seen^ 
however^ that zero is to b6 otherwise defined^ and has to be excluded before (2) 
becomes true ; while infinity is not a magnitude of the kind in question at all^ but 
merely a piece of mathematical shorthand. (Not infinity in general^ that is^ but 
infinite magnitude in the cases we are diseussing.) 



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ISl, 182] Infinity, the Infinitesimal, and Continuity 191 

magnitude would have a smaller difference from A than B has. Thus 
if (6) is universally true, (a) must also be true ; and conversely, if {a) is 
true, and if the series of magnitudes be such that equal stretches cor- 
respond to equal distances, then (h) is true of the distances between the 
magnitudes considered. We might rest content with the reduction of 
{a) to {h\ and proceed to the proof of (Jb) ; but it seems worth while 
to offer a direct proof, such as presumably the finitist philosopher has in 
his mind. 

Between A and B there is a certain number of magnitudes, unless A 
and B are consecutive. The intermediate magnitudes all have order, so 
that in passing from A to B all the intermediate magnitudes would 
be met with. In such an enumeration, there must be some magnitude 
which comes next after any magnitude C; or, to put the matter other- 
wise, since the enumeration has to begin, it must begin somewhere, and 
the term with which it begins must be the magnitude next to A. If 
this were not the case, there would be no definite series ; for if all the 
terms have an order, some of them must be consecutive. 

In the above argument, what is important is its dependence upon 
number. The whole argument turns upon the principle by which infinite 
number is shown to be self-contradictory, namely : A given collection 
of many terms must contain some finite number of terms. We say : All 
the magnitudes between A and B form a given collection. If there 
are no such magnitudes, ^ and B are consecutive, and the question 
is decided. If there are such magnitudes, there must be a finite 
number of them, say n. Since they form a series, there is a definite 
way of assigning to them the ordinal numbers from 1 to n. The mth 
and (//z+l)th are then consecutive. 

If the axiom in italics be denied, the whole argument collapses ; and 
this, we shall find, is also the case as regards (h) and (c). 

Q)) The proof here is precisely similar to the proof of {a). If there 
are no magnitudes less than A, then A is the least of its kind, and the 
question is decided. If there a;re any, they form a definite collection, 
and therefore (by our axiom) have a finite number, say n. Since they 
form a series, ordinal numbers may be assigned to them growing higher 
as the magnitudes become more distant from A. Thus the nth magni- 
tude is the smallest of its kind. 

{c) The proof here is obtained as in (&), by considering the collection 
of magnitudes greater than A. Thus everything depends upon our 
axiom, without which no case can be made out against continuity, or 
against the absence of a greatest and least magnitude. 

As regards the axiom itself, it will be seen that it has no particular 
reference to quantity, and at first sight it might seem to have no 
reference to order. But the word j^mz^^, which occurs in it, requires 
definition ; and this definition, in the form suited to the present dis* 
cussion, has, we shall find, an essential reference to order. 



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192 Quantity [chap, xxiii 

183. Of all the philosophers who have inveighed against infinite 
number, I doubt whether there is one who has known the difference 
between finite and infinite numbers. The difference is simply this. 
Finite numbers obey the law of mathematical induction ; infinite 
numbers do not. That is to say, given any number n^ if n belongs 
to every class s to which belongs, and to which belongs also the 
number next after any number which is an ^5 then n is finite ; if not, 
not. It is in this alone^ and in its consequences, that finite and infinite 
numbers differ*. 

^ The principle may be otherwise stated thus : If every proposition 
which holds concerning 0, and also holds concerning the immediate 
successor of every number of which it holds, holds concerning the number 
71, then n is finite ; if not, not. This is the precise sense of what may be 
popularly expressed by saying that every finite number can be reached 
from by successive steps, or by successive additions of 1. This is the 
principle which the philosopher must be held to lay down as obviously 
applicable to all numbers, though he will have to admit that the more 
precisely his principle is stated, the less obvious it becomes. 

184. It may be worth while to show exactly how mathematical 
induction enters into the above proofs. Let us take the proof of (^), 
and suppose there are n magnitudes between A and B. Then to begin 
with, we supposed these magnitudes capable of enumeration, i.e. of an 
order in which there are consecutive terms and a first term, and a term 
immediately preceding any term except the first. This property pre- 
supposes mathematical induction, and was in fact the very property in 
dispute. Hence we must not presuppose the possibility of enumeration, 
which would be a petitio principii. But to come to the kernel of the 
argument : we supposed that, in any series, there must be a definite way 
of assigning ordinal numbers to the terms. This property belongs to 
a series of one term, and belongs to every series having m -+■ 1 terms, 
if it belongs to every series having m terms. Hence, by mathematical 
induction, it belongs to all series having a finite number of terms. But 
if it be allowed that the number of terms may not be finite, the whole 
argument collapses. 

As regards (b) and (c), the argument is similar. Every series having 
a finite number of terms can be shown by mathematical induction to 
have a first and last term ; but no way exists of proving this concerning 
other series, or of proving that all series are finite. Mathematical 
induction, in short, like the axiom of parallels, is useful and convenient 
in its proper place ; but to suppose it always true is to yield to the 



■^ It must^ however^ be mentioned that one of these consequences gives a logical 
difference between finite and infinite numbers^ which may be taken as an inde- 
pendent definition. This has been already explained in Part 11^ Chap, xiii^ and will 
be further discussed in Part V. 



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183-185] Infinity, the Infinitesimal, and Continuity 193 

tyranny of mere prejudice. The philosopher's finitist arguments, there- 
fore, rest on a principle of which he is ignorant, which there is no reason 
to affirm, and every reason to deny. With this conclusion, the apparent 
antinomies may be considered solved. 

185. It remains to consider what kinds of magnitude satisfy the 
propositions (1), (2), (3). There is no general principle from which 
these can be proved or disproved, but there are certainly cases where 
they are true, and others where they are false. It is generally held by 
philosophers that numbers are essentially discrete, while magnitudes are 
essentially continuous. This we shall find to be not the case. Real 
numbers possess the most complete continuity known, while many kinds 
of magnitude possess no continuity at all. The word continuity has 
many meanings, but in mathematics it has only two — one old, the other 
new. For present purposes the old meaning will suffice. I therefore 
set up, for the present, the following definition : 

Continuity applies to series (and only to series) whenever these are 
such that there is a term between any two given terms *. Whatever is 
not a series, or a compound of series, or whatever is a series not fulfilling 
the above condition, is discontinuous. 

Thus the series of rational numbers is continuous, for the arithmetic 
mean of two of them is always a third rational number between the two. 
The letters of the alphabet are not continuous. 

We have seen that any two terms in a series have a distance, or a 
stretch which has magnitude. Since there are certainly discrete series 
{e.g. the alphabet), there are certainly discrete magnitudes, namely, the 
distances or the stretches of terms in discrete series. The distance 
between the letters A and C is greater than that between the letters 
A and B, but there is no magnitude which is greater than one of these 
and less than the other. In this case, there is also a greatest possible 
and a least possible distance, so that all three propositions (1), (S), (3) 
fail. It must not be supposed, however, that the three propositions 
have any necessary connection. In the case of the integers, for example, 
there are consecutive distances, and there is a least possible distance, 
namely, that between consecutive integers, but there is no greatest 
possible distance. Thus (3) is true, while (1) and (S) are false. In 
the case of the series of notes, or of colours of the rainbow, the series 
has a beginning and end, so that there is a greatest distance ; but there 
is no least distance, and there is a term between any two. Thus (1) 
and (2) are true, while (3) is false. Or again, if we take the series 
composed of zero and the fractions having one for numerator, there is a 

* The objection to this definition (as we shall see in Part V) is^ that it does not 
give the usual properties of the existence of limits to convergent series which are 
commonly associated with continuity. Series of the above kind will be called 
compact^ except in the present discussion. 

R. 13 



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194 Quantity [chap, xxiil 

greatest distance, but no least distance, though the series is discrete. 
Thus (S) is true, while (1) and (3) are false. And other combinations 
might be obtained from other series. 

Thus the three propositions (1), (S), (3), have no necessary connection, 
and all of them, or any selection, may be false as applied to any given 
kind of magnitude. We cannot hope, therefore, to prove their truth 
from the nature of magnitude. If they are ever to be true, this must be 
proved independently, or discovered by mere inspection in each particular 
case. That they are sometimes true, appears from a consideration of 
the distances between terms of the number-continuum or of the rational 
numbers. Either of these series is continuous in the above sense, and 
has no first or last term (when zero is excluded). Hence its distances 
or stretches fulfil all three conditions. The same might be inferred 
from space and time, but I do not wish to anticipate what is to be said 
of these. Quantities of divisibility do not fulfil these conditions when 
the wholes which are divisible consist of a finite number of indivisible 
parts. But where the number of parts is infinite in a whole class of 
differing magnitudes, all three conditions are satisfied, as appears from 
the properties of the number-continuum. 

We thus see that the problems of infinity and continuity have no 
essential connection with quantity, but are due, where magnitudes 
present them at all, to characteristics depending upon number and 
order. Hence the discussion of these problems can only be undertaken 
after the pure theory of order has been set forth*. To do this -will 
be the aim of the following Part. 

186. We may now sum up the results obtained in Part III. In 
Chapter xix we determined to define a magnitude as whatever is either 
greater or less than something else. We found that magnitude has no 
necessary connection with divisibility, and that greater and less are inde- 
finable. Every magnitude, we saw, has a certain relation — analogous to, 
but not identical with, that of inclusion in a class — to a certain quality 
or relation ; and this fact is expressed by saying that the magnitude 
in question is a magnitude of that quality or relation. We defined a 
quantity as a particular contained under a magnitude, i.e. as the complex 
consisting of a magnitude with a certain spatio-temporal position, or with 
a pair of terms between which it is a relation. We decided, by means of 
a general principle concerning transitive symmetrical relations, that 
it is impossible to content ourselves with quantities, and deny the 
further abstraction involved in magnitudes; that equality is not a direct 
relation between quantities, but consists in being particularizations of 
the same magnitude. Thus equal quantities are instances of the same 



■^ Cf. Couturat^ ^^ Sur la Definition du Continu," Eevue de Metaphysique et de 
Morale, 1900. 



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185, 186] Infinity, the Infinitesimal, and Continuity 195 

magnitude. Similarly greater and less are not direct relations between 
quantities, but between magnitudes : quantities are only greater and 
less in virtue of being instances of greater and less magnitudes. Any 
two magnitudes which are of the same quality or relation are one 
greater, the other less ; and greater and less are asymmetrical transitive 
relations. 

Among the terms which have magnitude are not only many qualities, 
but also asymmetrical relations by which certain kinds of series are 
constituted. These may be called distances. When there are distances 
in a series, any two terms of the series have a distance, which is the same 
as, greater than, or less than, the distance of any two other terms in the 
series. Another peculiar class of magnitudes discussed in Chapter xx is 
constituted by the degrees of divisibility of different wholes. This, we 
found, is the only case in which quantities are divisible, while there is no 
instance of divisible magnitudes. 

Numerical measurement, which was discussed in Chapter xxi, required, 
owing to the decision that most quantities and all magnitudes are in- 
divisible, a somewhat unusual treatment. The problem lies, we found, 
in establishing a one-one relation between numbers and the magnitudes 
of the kind to be measured. On certain metaphysical hypotheses (which 
were neither accepted nor rejected), this was found to be always theo- 
retically possible as regards existents actual or possible, though often 
not practically feasible or important. In regard to two classes of 
magnitudes, namely divisibilities and distances, measurement was found 
to proceed from a very natural convention, which defines what is 
meant by saying (what can never have the simple sense which it has in 
connection with finite wholes and parts) that one such magnitude is 
double of, or n times, another. The relation of distance to stretch 
was discussed, and it was found that, apart from a special axiom to 
that effect, there was no a priori reason for regarding equal distances as 
corresponding to equal stretches. 

In Chapter xxii we discussed the definition of zero. The problem 
of zero was found to have no connection with that of the infini- 
tesimal, being in fact closely related to the purely logical problem 
as to the nature of negation. We decided that, just as there are the 
distinct logical and arithmetical negations, so there is a third funda- 
mental kind, the quantitative negation ; but that this is negation of 
that quality or relation of which the magnitudes are, not of magnitude 
of that quality or relation. Hence we were able to regard zero as one 
among the magnitudes contained in a kind of magnitude, and to dis- 
tinguish the zeroes of different kinds. We showed also that quantitative 
negation is connected with logical negation by the fact that there cannot 
be any quantities whose magnitude is zero. 

In the present Chapter the problems of continuity, the infinite, and 
the infinitesimal, were shown to belong, not specially to the theory of 

13—2 



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196 Quantity [chap, xxiii 

quantity, but to those of number and order. It was shown that, though 
there are kinds of magnitude in which there is no greatest and no least 
magnitude, this fact does not require us to admit infinite or infinitesimal 
magnitudes ; and that there is no contradiction in supposing a kind of 
magnitudes to form a series in which there is a term between any two, 
and in which, consequently, there is no term consecutive to a given term. 
The supposed contradiction was shown to result from an undue use of 
mathematical induction — a principle, the full discussion of which pre- 
supposes the philosophy of order. 



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PAET lY. 

ORDEE. 



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CHAPTER XXIV. 



THE GENESIS OF SERIES. 



187. The notion of order or series is one with whicli, in connection 
with distance, and with the order of magnitude, we have already 
had to deal. The discussion of continuity in the last chapter of 
Part III showed us that this is properly an ordinal notion, and 
prepared us for the fundamental importance of order. It is now high 
time to examine this concept on its own account. The importanc^e of 
order, from a purely mathematical standpoint, has been immeasurably 
increased by many modern developments. Dedekind, Cantor, and Peano 
have shown how to base all Arithmetic and Analysis upon series of a 
certain kind — i.e, upon those properties of finite numbers in virtue 
of which they form what I shall call a progression. Irrationals are 
defined (as we shall see) entirely by the help of order; and a new 
class of transfinite ordinals is introduced, by which the most important 
and interesting results are obtained. In Geometry, von Staudt's quadri- 
lateral construction and Pieri's work on Projective Geometry have shown 
how to give points, lines, and planes an order independent of metrical 
considerations and of quantity ; while descriptive Geometry proves that 
a very large part of Geometry demands only the possibility of serial 
arrangement. Moreover the whole philosophy of space and time depends 
upon the view we take of order. Thus a discussion of order, which 
is lacking in the current philosophies, has become essential to any 
understanding of the foundations of mathematics. 

188. The notion of order is more complex than any hitherto 
analyzed. Two terms cannot have an order, and even three cannot 
have a cyclic order. Owing to this complexity, the logical analysis 
of order presents considerable diflBculties. I shall therefore approach 
the problem gradually, considering, in this chapter, the circumstances 
under which order arises, and reserving for the second chapter the 
discussion as to what order really is. This analysis will raise several 
fundamental points in general logic, which will demand considerable 
discussion of an almost purely philosophical nature. From this I shall 
pass to more mathematical topics, such as the types of series and 



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200 Order [chap, xxiv 

the ordinal definition of numbers, thus gradually preparing the way for 
the discussion of infinity and continuity in the following Part. 

There are two different ways in which order may arise, though we 
shall find in the end that the second way is reducible to the first. In 
the first, what may be called the ordinal element consists of three terms 
a, 5, c, one of which (6 say) is between the other two. This happens 
whenever there is a relation of <^ to 6 and of h to c, which is not a 
relation of 5 to <2, of c to 5, or of c to a. This is the definition, or 
better perhaps, the necessary and suflScient condition, of the proposition 
" 6 is between a and c^ But there are other cases of order where, at 
first sight, the above conditions are not satisfied, and where between 
is not obviously applicable. These are cases where we have four terms 
a, 6, 6', d^ as the ordinal element, of which we can say that a and c are 
separated by b and d. This relation is more complicated, but the 
following seems to characterize it : a and c are separated from b and d^ 
when there is an asymmetrical relation which holds between a and 5, 
b and 6*, c and d^ or between a and d^ d and c^ c and b^ or between 
c and d^ d and a^ a and b ; while if we have the first case, the same 
relation must hold either between d and a, or else between both a 
and c, and a and d\ with similar assumptions for the other two cases*. 
(No further special assumption is required as to the relation between 
a and c or between b and d-^ it is the absence of such an assumption 
which prevents our reducing this case to the former in a simple manner.) 
There are cases — notably where our series is closed — in which it seems 
formallv impossible to reduce this second case to the first, though this 
appearance, as we shall see, is in part deceptive. We have to show, 
in the present chapter, the principal ways in which series, arise from 
collections of such ordinal elements. 

Although two terms alone cannot have an order, we must not 
assume that order is possible except where there are relations between 
two terms. In all series, we shall find, there are asymmetrical relations 
between two terms. But an asymmetrical relation of which there is 
only one instance does not constitute order. We require at least two 
instances for between^ and at least three for separation of pairs. Thus 
although order is a relation between three or four terms, it is only 
possible where there are other relations which hold between pairs of 
terms. These relations may be of various kinds, giving different ways 
of generating series. I shall now enumerate the principal ways with 
which I am acquainted. 

189. (1) The simplest method of generating a series is as follows. 
Let there be a collection of terms, finite or infinite, such that every 
term (with the possible exception of a single one) has to one and only 

■^ This gives a sufficient but not a necessary condition for the separation 
of couples. 



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188, 189] The Genesis of Series 201 

one other term of the collection a certain asymmetrical relation (which 
must of course be intransitive), and that every term (with again one 
possible exception, which must not be the same as the term formerly 
excepted) has also to one and only one other term of the collection 
the relation which is the converse of the former one*. Further, let 
it be assumed that, if a has the first relation to 6, and 5 to c, then c 
does not have the first relation to a. Then every term of the collection 
except the two peculiar terms has one relation to a second term, and 
the converse relation to a third, while these terms themselves do not 
have to each other either of the relations in question. Consequently, 
by the definition of between^ our first term is between our second and 
third terms. The term to which a given term has one of the two 
relations in question is called newt after the given term ; the term to 
which the given term has the converse relation is called next before 
the given term. Two terms between which the relations in question 
hold are called consecutive. The exceptional terms (when they exist) 
are not between any pair of terms • they are called the two ends of 
the series, or one is called the beginning and the other the end. The 
existence of the one does not imply that of the other — for example 
the natural numbers have a beginning but no end — and neither need 
exist — for example, the positive and negative integers together have 

neither t. 

The above method may perhaps become clear by a formal exhibition. 

Let R be one of our relations, and let its converse be denoted by R\, 
Then if e be any term of our set," there are two terms d^f such that 
e R d^ e Rf i.e. such that d R e^ e R f. Since each term only has the 
relation R 'to one other, we cannot have d Rf\ and it was one of 
the initial assumptions that we were not to have y R d. Hence e is 
between d andy |. If a be a term which has only the relation i?, then 
obviously a is not between any pair of terms. We may extend the 
notion of between by defining that, if c be between b and d^ and d 
between c and ^, then c or d will be said to be also between b and e. 
In this way, unless we either reach an end or come back to the term 
with which we started, we can find any number of terms between which 
and b the term c will lie. But if the total number of terms be not 
less than seven, we cannot show in this way that of any three terms 
one must be between the other two, since the collection may consist 

* The converse of a relation is the relation which must hold between 1/ and 00 
when the given relation holds between w and y. 

t The above is the only method of generating series given by Bolzano^ *^^Para- 
doxlen des Unendlichen_," § 7. 

X This is the notation adopted by Professor Schroder. 

§ The denial of 6? i?y* is only necessary to this special method^ but the denial ot 
fR d is essential to the definition of between. 



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202 Order [chap, xxiy 

of two distinct series, of which, if the collection is finite, one at least 
must be closed, in order to avoid more than two ends. 

This remark shows that, if the above method is to give a single 
series, to which any term of om^ collection is to belong, we need a 
further condition, which may be expressed by saying that the collection 
must be connected. We shall find means hereafter of expressing this 
condition without reference to number, but for the present we may 
content ourselves by saying that our collection is connected when, given 
any two of its terms, there is a certain finite number (not necessarily 
unique) of steps from one term to the next, by which we can pass 
from one of our two terms to the other. When this condition is 
fulfilled, we are assured that, of any three terms of our collection, one 
must be between the other two. 

Assuming now that our collection is connected, and therefore forms 
a single series, four cases may arise : {a) our series may have two ends, 
{h) it may have one end, (c) it may have no end and be open, {d) it may 
have no end and be closed. Concerning («), it is to be observed that 
our series must be finite. For, taking the two ends, since the collection 
is connected, there is some finite number 71 of steps which will take 
us from one end to the other, and hence n^\ is the number of terms 
of the series. Every term except the two ends is between them, and 
neither of them is between any other pair of terms. In case (&), on 
the other hand, our collection must be infinite, and this would hold 
even if it were not connected. For suppose the end which exists to 
have the relation i?, but not R. Then every other term of the collection 
has both relations, and can never have both to the same term, since R 
is asymmetrical. Hence the term to which (say) e has the relation R is 
not that to which it had the relation R^ but is either some new term, 
or one of ^""s predecessors. Now it cannot be the end-term a^ since 

a does not have the relation R to any term. Nor can it be any term 
which can be reached by successive steps from a without passing 
through ^, for if it were, this term would have two predecessors, 
contrary to the hypothesis that J? is a one-one relation. Hence, if 
h be any term which can be reached by successive steps from a, 
A; has a successor which is not a or any of the terms between a 
and A; ; and hence the collection is infinite, whether it be connected 
or not. In case (c), the collection must again be infinite. For here, 
by hypothesis, the series is open — i.^., starting from any term ^, no 
number of steps in either direction brings us back to e. And there 
cannot be a finite limit to the number of possible steps, since, if there 
were, the series would have an end. Here again, it is not necessary to 
suppose the series connected. In case (d\ on the contrary, we must 
assume connection. By saying that the series is closed, we mean that 
there exists some number n of steps by which, starting from a certain 



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189, 190] The Genesis of Series 203 

term a^ we shall be brought back to a. In this case, n is the number 
of terms, and it makes no difference with which term we start. In this 
case, between is not definite except where three terms are consecutive, 
and the series contains more than three terms. Otherwise, we need the 
more complicated relation of separation. 

190. (2) The above method, as we have seen, will give either open 
or closed series, but only such as have consecutive terms. The second 
method, which is now to be discussed, will give series in which there 
are no consecutive terms, but will not give closed series*. In this 
method we have a transitive asymmetrical relation P, and a collection 
of terms any two of which are such that either xPy or yPx. When 
these conditions are satisfied our terms necessarily form a single series. 
Since the relation is asymmetrical, we can distinguish xPy from yPx^ 
and the two cannot both subsist f. Since P is transitive, ocPy and «/P^ 
involve xPz. It follows that P is also asymmetrical and transitive J. 
Thus with respect to any term x of our collection, all other terms of 
the collection fall into two classes, those for which xPy^ and those for 
which zPx. Calling these two classes m-x and irx respectively, we see 
that, owing to the transitiveness of P, if y belongs to the class tt/t, 
iry is contained in ttx ; and if z belongs to the class irx^ irz is contained 
in TTX. Taking now two terms x, y, for which xPy, all other terms fall 
into three classes : (1) Those belonging to ttx, and therefore to iry ; 
(^) those belonging to tt?/, and therefore to ttx ; (3) those belonging to 
TTX but not to TT?/. If z be of the first class, we have zPx, zPy ; if i^ be 
of the second, xPv and yPv ; if w be of the third, xPw and wPy. The 
case yPii and uPx is excluded : for xPy^ yPu imply xPu^ which is in- 
consistent with uPx, Thus we have, in the three cases, (1) ^ is between 
z and y ; (2) y is between x and t; ; (3) w \% between x and y. Hence 
any three terms of our collection are such that one is between the other 
two, and the whole collection forms a single series. If the class (3) 
contains no terms, x and y are said to be consecutive ; but many rela- 
tions P can be assigned, for which there are always terms in the class (3). 
If for example P be before^ and our collection be the moments in a 

^ The following method is the only one given by Vivanti in the Formulaire de 
MathematiqueSy (1895)^ vi, § 2, No. 7 ; also by Oilman^ '^'^On the properties of a one- 
dimensional manifold/' Mind, N.S. Vol. i. We shall find that it is general in 
a sense in which none of our other methods are so. 

t I use the term asymmetrical as the contrary^ rather than the contradictory^ 
of symmetrical. If ^Py, and the relation is symmetrical^ we have always yPa? ; if 
asymmetrical^ we never have ?/P^. Some relations — e.g. logical implication — are 
neither symmetrical nor asymmetrical. Instead of assuming P to be asymmetrical^ 
we may make the equivalent assumption that it is what Professor Peirce calls an 
aliorelative, i.e. a relation which no term has to itself. (This assumption is not 
equivalent to asymmetry in general^ but only when combined with transitiveness.) 

X P may be read precedes, and P may be read follows, provided no temporal or 
spatial ideas are allowed to intrude themselves. 



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204 Order [chap, xxiy 

certain interval^ or in all time, there is a moment between any two of 
our collection. Similarly in the case of the magnitudes which, in the 
last chapter of Part III, we called continuous. There is nothing in 
the present method, as there was in the first, to show that there must 
be consecutive terms, unless the total number of terms in our collection 
be finite. On the other hand, the present method will not allow closed 
series ; for owing to the transitiveness of the relation P, if the series 
were closed, and x were any one of its terms, we should have ocPx^ which 
is impossible because P is asymmetrical. Thus in a closed series, the 
generating relation can never be transitive*. As in the former method, 
the series may have two ends, or one, or none. In the first case only, 
it may be finite ; but even in this case it may be infinite, and in the 
other two cases it must be so. 

191. (3) A series may be generated by means of distances, as was 
already partially explained in Part III, and as we shall see more fully 
hereafter. In this case, starting with a certain term x^ we are to have 
relations, which are magnitudes, between oc and a number of other terms 

y^ z According as these relations are greater or less, we can order 

the corresponding terms. If there are no similar relations between the 
remaining terms ?/, <^, ..., we require nothing further. But if these 
have relations which are magnitudes of the same kind, certain axioms 
are necessary to insure that the order may be independent of the 
particular term from which we start. Denoting by xz the distance of 
X and 2, if xz is less than xtv^ we must have yz less than yw. A con- 
sequence, which did not follow when x was the only term that had 
a distance, is that the distances must be asymmetrical relations, and 
those which have one sense must be considered less than zero. For 
" xz is less than xko'''' must involve " ivz is less than wkd^'''' i.e. wz is less 
than 0. In this way the present case is practically reduced to the 
second ; for every pair of terms x^ y will be such that xy is less than 
or else xy is greater than ; and we may put in the first case yPx^ 
in the second xPy. But we require one further axiom in order that 
the arrangement may be thus effected unambiguously, li xz = yiv., and 
Zto'= xy^ ziv and tv' must be the same point. With this further axiom, 
the reduction to case (2) becomes complete. 

192. (4) Cases of triangular relations are capable of giving rise to 
order. Let there be a relation R which holds between y and (x, z\ 
between z and (?/, u)^ between u and (^, w)^ and so on. Between is itself 
such a relation, and this might therefore seem the most direct and 
natural way of generating order. We should say, in such a case," that y 
is between x and 2, when the relation R holds between y and the couple 
x^ z. We should need assumptions concerning R which should show 
that, if y is between x and z^ and z between y and w^ then y and z are 

* For more precise statements_, see Chap, xxviii. 



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190-194] The Genesis of Series 205 

each between oc and w. That is, if we have yR {oo^ z)^ zR (?/, w)^ we must 
have yR {x^ w) and zR {oc^ iv). This is a kind of three -term transitiveness. 
Also if y be between x and %v^ and z between y and iv^ then z must be 
between x and w^ and ?/ between ^' and z : that is, if yR {x^ iv) and 
zR{y^ %v)^ then zR{x^ iv) and yR{x^ z). Also yR{x^ z) must be equi- 
valent to yR(z^ ^)*. AVith these assumptions, an unambiguous order 
will be generated among any number of terms such that any triad has 
the relation R. Whether such a state of things can ever be incapable of 
further analysis, is a question which I leave for the next chapter. 

193. (5) We have found hitherto no way of generating closed 
continuous series. There are, however, instances of such series, e.g. 
angles, the elliptic straight line, the complex numbers with a given 
modulus. It is therefore necessary to have some theory which allows of 
their possibility. In the case where our terms are asymmetrical relations, 
as straight lines are, or are correlated uniquely and reciprocally with 
such relations, the following theory will effect this object. In other 
cases, the sixth method (below) seems adequate to the end in view. 

Let ti', y^ z... be a set of asymmetrical relations, and let R be an 
asymmetrical relation which holds between any two x^ y or ?/, x except 
when y is the converse relation to x. Also let R be such that, if it holds 
between x and y^ it holds between y and the converse of x ; and if x be 
any term of the collection, let all the terms to which x has either of the 
relations /^, R be terms of the collection. All these conditions are 
satisfied by angles, and whenever they are satisfied, the resulting series is 
closed. For xRy implies yRx^ and hence xRy^ and thence yRx ; so 
that by means of relations R it is possible to travel from x back to x. 
Also there is nothing in the definition to show that our series cannot be 
continuous. Since it is closed, we cannot apply universally the notion of 
hetioeen ; but the notion of separation can be always applied. The 
reason why it is necessary to suppose that our terms either are, or are 
correlated with, asymmetrical relations, is, that such series often have 
antipodes, opposite terms as they may be called ; and that the notion of 
opposite seems to be essentially bound up with that of the converse of an 
asymmetrical relation. 

194. (6) In the same way in which, in (4), we showed how to 
construct a series by relations of hetiveen^ we can construct a series 
directly by four-term relations of separation. For this purpose, as 
before, certain axioms are necessary. The following five axioms have 
been shown by Vailatif to be sufficient, and by Padoa to possess ordered 
independence, i.e. to be such that none can be deduced from its pre- 
decessors j. Denoting "a and h separate c from cV by ah\\cd^ we must 
have : 

■^ See Peaiio^ I Principii di G.eometria^ Turing, 1889^ Axioms viii, ix^ x^ xi. 
+ Rwista di Mateonatica, y, pp. 76^ 183. % Ibid. p. 185. 



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206 Order [chap, xxiy 

(a) ah^cd is equivalent to cd\\ah'^ 

(/3) ah^cd is equivalent to ab\\dc'^ 

(7) «5 II C6? excludes «c || M ; 

(S) For any four terms of our collection, we must have ah\^cd^ or 
ac\\ bd, or ad\\bc; 

(e) If tt& II C6Z, and ac \\ be, then ac \\de. 

By means of these five assumptions, our terms a, b, c, d, e... acquire 
an unambiguous order, in which we start from a relation between two 
pairs of terms, which is undefined except to the extent to which the 
above assumptions define it. The further consideration of this case, as 
generally of the relation of separation, I postpone to a later stage. 

The above six methods of generating series are the principal ones 
with which I am acquainted, and all other methods, so far as I know, 
are reducible to one of these six. The last alone gives a method of 
generating closed continuous series whose terms neither are, nor are 
correlated with, asymmetrical relations*. This last method should there- 
fore be applied in projective and elliptic Geometry, where the correlation 
of the points on a line with the lines through a point appears to be 
logically subsequent to the order of the points on a line. But before we 
can decide whether these six methods (especially the fourth and sixth) 
are irreducible and independent, we must discuss (what has not hitherto 
been analyzed) the meaning of order, and the logical constituents (if any) 
of which this meaning is compounded. This will be done in the following 
chapter. 

■^ See Chap, xxviii. 



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CHAPTEE XXV. 



THE MEANING OF ORDER. 

195. We have now seen under what circumstances there is an order 
among a set of terms, and by this means we have acquired a certain 
inductive famiharity with the nature of order. But we have not yet 
faced the question : What is order ? This is a difficult question, and 
one upon which, so far as I know, nothing at all has been Avritten. All 
the authors with whom I am acquainted are content to exhibit the 
genesis of order ; and since most of them give only one of the six 
methods enumerated in Chapter xxiv, it is easy for them to confound the 
genesis of order with its nature. This confusion is rendered evident to 
us by the multiplicity of the above methods ; for it is evident that we 
mean by order something perfectly definite, which, being generated 
equally in all our six cases, is clearly distinct from each and all of the 
ways in which it may be generated, unless one of these ways should turn 
out to be fundamental, and the others to be reducible to it. To elicit 
this common element in all series, and to broach the logical discussions 
connected with it, is the purpose of the present chapter. This discussion 
is of purely philosophical interest, and might be wholly omitted in a 
mathematical treatment of the subject. 

In order to approach the subject gradually, let us separate, the 
discussion of hettveen from that of separation of couples. When we have 
decided upon the nature of each of these separately, it will be time to 
combine them, and examine what it is that both have in common. 
I shall begin with between^ as being the simpler of the two. 

196. Between may be characterized (as in Chapter xxiv) as a relation 
of one term y to two others x and ^, which holds w^henever oo has to ?/, and 
y has to ^, some relation which y does not have to ^, nor z to «/, nor z to x^. 

^ The condition that z does not have to w the relation in question is comparatively 
inessential_, being only required in order that^ if y be between oo and z, we may not 
have 00 between y and z, or z between oo and y. If we are willing to allow that in 
such caseS;, for example^ as the angles of a triangle^ each is between the other two_, 
we may drop the condition in question altogether. The other four conditions^ on 
the contrary^ seem more essential. 



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208 Order [chap, xxv 

These conditions are undoubtedly sufficient for betweenness, but it may 
be questioned whether they are necessary. Several possible opinions 
must be distinguished in this respect. (1) We may hold that the above 
conditions give the very meaning of between, that they constitute an 
actual analysis of it, and not merely a set of conditions insuring its 
presence. (2) We may hold that hetuDeen is not a relation of the terms 
c^', ?/, z at all, but a relation of the relation of y to x to that of y to z^ 
namely the relation of difference of sense. (3) We may hold that 
hetiveen is an indefinable notion, like greater and less ; that the above 
conditions allow us to infer that y is between x and z^ but that there 
may be other circumstances under which this occurs, and even that it 
may occur without involving any relation except diversity among the 
pairs {oCj y)^ (?/, z\ (^, z). In order to decide between these theories, it 
will be well to develop each in turn. 

197. (1) In this theory, we define "y is between x and ^" to mean : 
" There is a relation R such that xRy^ yRz but not yRx^ zRy '' ; and it 
remains a question whether we are to add " not zRx."" We will suppose 
to begin with that this addition is not made. The following propositions 
will be generally admitted to be self-evident : (a) If y be between x and z, 
and z between y and to, then y is between x and tv ; {(3) if y be between 
X and z, and to between x and y^ then y is between w and z. For brevity, 
let us express "y is between x and z'''' by the symbol xyz. Then our two 
propositions are : (a) xyz and yzw imply xyz^ ; (/3) xyz and xzvy imply 
z^yz. We must add that the relation of hettveen is symmetrical so far as 
the extremes are concerned : i.e. xyz implies zyx. This condition follows 
directly from our definition. With regard to the axioms (a) and (/9), it 
is to be observed that betzveen, on our present view, is always relative to 
some relation R, and that the axioms are only assumed to hold when it 
is the same relation R that is in question in both the premisses. Let us 
see whether these axioms are consequences of our definition. For this 
purpose, let us write R for not -R. 

xyz means xRy^ yRz, yRx, zRy. 

yzzo means yRz, zRzv, zRy, zoRz. 

Thus yzzo only adds to xyz the two conditions zRzo, zvRz. If R is 
transitive, these conditions insure xyzv ; if not, not. Now we have seen 
that some series are generated by one-one relations J?, which are not 
transitive. In these cases, however, denoting by R^ the relation between 
X and z implied by xRy, yRz, and so on for higher powers, we can 
substitute a transitive relation R' for R, where R' means "some positive 
power of J?.^' In this way, if xyz holds for a relation which is some 
definite power of R, then xyz holds for R\ provided only that no positive 

power of R is equivalent to R. For, in this latter event, we should 
have yR^x whenever xR'y, and R' could not be substituted for R in the 
explanation of xyz. Now this condition, that the converse of R is not 



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196, 197] The meaning of Order 209 

to be a positive power of J?, is equivalent to the condition that our 

series is not to be closed. For \i R = R^\ then RR = J?'^+i ; but since R 

is a one-one relation, RR implies the relation of identity. Thus n-\-l 
steps bring us back from <r to a,', and our series is a closed series of 
71 + 1 terms. Now we have agreed already that between is not properly 

applicable to closed series. Hence this condition, that R is not to 
be a power of i?, imposes only such restrictions upon our axiom (a) as 
we should expect it to be subject to. 

With regard to (/3), we have 

ojt/z = ojRy . yRz . i/Rx . zRi/. 

xwy = ocRio . wRy . loRcc . yRw. 

The case contemplated by this axiom is only possible if R be not 
a one-one relation, since we have xRy and xRw. The deduction noyz 
is here an immediate consequence of the definition, without the need of 
any further conditions. 

It remains to examine whether we can dispense with the condition 
zRx in the definition of hetiveen. If we suppose i? to be a one-one 
relation, and zRx to be satisfied, we shall have 

wyz = xRy . yRz . zRy.yRoc^ 

and we have further by hypothesis zRx^ and since R is one -one, and 
xRy^ we have xRz. Hence, in virtue of the definition, we have yzx ; 
and similarly we shall obtain zxy. If we now adhere to our axiom (a), 
we shall have xzx^ which is impossible ; for it is certainly part of the 
meaning of hetween that the three terms in the relation should be 
different, and it is impossible that a term should be between x and x. 
Thus we must either insert our condition zRx^ or we must set up the 
new condition in the definition, that x and z are to be different. (It 
should be observed that our definition implies that x is different from y 
and y from z ; for if not, xRy would involve yRx^ and yRz would 
involve zRy.) It would seem preferable to insert the condition that x 
and z are to be different : for this is in any case necessary, and is not 
implied by .zRx, This condition must then be added to our axiom (a) ; 
xyz and yzxv are to imply xyw^ unless x and tv are identical. In axiom 
(/3), this addition is not necessary, since it is implied in the premisses. 
Thus the condition zRx is not necessary, if we are willing to admit that 
xyz is compatible with yzx- — an admission which such cases as the 
angles of a triangle render possible. Or we may insert, in place of 
zRx^ the condition which we found necessary before to the universal 
validity of our axiom (a), namely that no power of R is to be equivalent 
to the converse of R : for if we have both xyz and yzx, we shall have (so 

far at least as x^y, z are concerned) R^ = R, i.e. if xRy and yRz, then 
E. 14 



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210 Order [chap, xxv 

zRx. This last course seems to be the best. Hence in all cases where 
our first instance of hetimen is defined by a one-one relation R^ we shall 
substitute the relation R\ which means "some positive power of iZ."' 
The relation R' is then transitive, and the condition that no positive 
power of R is to be equivalent to R is equivalent to the condition that 
jRMs to be asymmetrical. Hence, finally, the whole matter is simplified 
into the following : 

To say that y is between do and z is equivalent to saying that there 
is some transitive asymmetrical relation which relates both oc and ?/, and 
y and z. 

This short and simple statement, as the above lengthy argument 
shows, contains neither more nor less than our original definition, to- 
gether with the emendations which we gradually found to be necessary. 
The question remains, however : Is this the meaning of betzoeen ? 

198. A negative instance can be at once established if we allow the 
phrase : J? is a relation between x and y. The phrase, as the reader will 
have observed, has been with difficulty excluded from the definitions of 
between^ which its introduction would have rendered at least verbally 
circular. The phrase may have none but a linguistic importance, or 
again it may point to a real insufficiency in the above definition. Let 
us examine the relation of a relation R to its terms x and y. In the 
first place, there certainly is such a relation. To be a term which has 
the relation R to some other term is certainly to have a relation to R^ 
a relation which we may express as '' belonging to the domain of R^ 
Thus if xRy^ x will belong to the domain of jR, and y to that of R. 
If we express this relation between x and i?, or between y and i?, by jEJ, 
we shall have xKR^ yER. If further we express the relation of R to R 
by /, we shall have RIR and RIR, Thus we have xER, yEIR. Now 
EI is by no means the converse of E^ and thus the above definition of 
between^ if for this reason only, does not apply ; also neither E nor EI 
is transitive. Thus our definition of between is wholly inapplicable to 
such a case. Now it may well be doubted whether between^ in this case, 
has at all the same meaning as in other cases. Certainly we do not in 
this way obtain series : x and y are not, in the same sense as i?, between 
R and other terms. Moreover, if we admit relations of a term to itself, 
we shall have to admit that such relations are between a term and 
itself, which we agreed to be impossible. Hence we may be tempted 
to regard the use of betweeri in this case as due to the linguistic accident 
that the relation is usually mentioned between the subject and the 
object, as in "A is the father of J5." On the other hand, it may be 
urged that a relation does have a very peculiar relation to the pair of 
terms which it relates, and that bettveen should denote a relation of one 
term to two others. To the objection concerning relations of a term 
to itself, it may be answered that such relations, in any system, con- 



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197-199] The meaning of Order 211 

stitute a grave logical difficulty; that they would, if possible, be denied 
philosophic validity ; and that even where the relation asserted is 
identity, there must be two identical terms, which are therefore not 
quite identical. As this raises a fundamental difficulty, which we cannot 
discuss here, it will be prudent to allow the answer to pass*. And it 
may be further urged that use of the same word in two connections^ 
points always to some analogy, the extent of which should be carefully 
indicated by those who deny that the meaning is the same in both 
cases ; and that the analogy here is certainly profounder than the mere 
order of words in a sentence, which is, in any case, far more variable 
in this respect than the phrase that a relation is between its terms. 
To these remarks, however, it may be retorted that the objector has 
himself indicated the precise extent of the analogy : the relation of a 
relation to its terms is a relation of one term to two others, just as 
between is, and this is what makes the two cases similar. This last 
retort is, I think, valid, and we may allow that the relation of a relation 
to its terms, though involving a most important logical problem, is 
not the same as the relation of between by which order is to be con- 
stituted. 

Nevertheless, the above definition of between^ though we shall be 
ultimately forced to accept it, seems, at first sight, scarcely adequate 
from a philosophical point of view. The reference to some asymmetrical 
relation is vague, and seems to require to be replaced by some phrase 
in which no such undefined relation appears, but only the terms and 
the betweenness. This brings us to the second of the above opinions 
concerning between. 

199. (2) Between., it may be said, is not a relation of three terms 
at all, but a relation of two relations, namely difference of sense. Now 
if we take this view, the first point to be observed is, that we require 
the two opposite relations, not merely in general, but as particularized 
by belonging to one and the same term. This distinction is already 
familiar from the case of magnitudes and quantities. Before and aftei 
in the abstract do not constitute between-, it is only when one and the 
same term is both before and after that between arises : this term is 
then between what it is before and what it is after. Hence there is 
a difficulty in the reduction of between to difference of sense. The par- 
ticularized relation is a logically puzzling entity, which in Part I (§ ^^^ 
we found it necessary to deny ; and it is not quite easy to distinguish 
a relation of two relations, particularized as belonging to the same term, 
from a relation of the term in question to two others. At the same 
time, great advantages are secured by this reduction. We get rid of 
the necessity for a triangular relation, to which many philosophers may 
object, and we assign a common element to all cases of between^ namely 
difference of sense, i.e. the difference between an asymmetrical relation 
and its converse. 

^ Cf. § 95. 

14—2 



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212 Order [chap, xxv 

200. The , question whether there can be an ultimate triangular 
relation is one whose actual solution is both difficult and unimportant, 
but whose precise statement is of very great importance. Philosophers 
seem usually to assume — though not, so far as I know, explicitly — that 
relations never have more than two terms ; and even such relations they 
reduce, by force or guile, to predications. Mathematicians, on the other 
hand, almost invariably speak of relations of many terms. We cannot, 
however, settle the question by a simple appeal to mathematical instances, 
for it remains a question whether these are, or are not, susceptible of 
analysis. Suppose, for example, that the projective plane has been 
defined as a relation of three points : the philosopher may always say 
that it should have been defined as a relation of a point and a line, 
or of two intersecting lines — a change which makes little or no mathe- 
matical difference. Let us see what is the precise meaning of the question. 
There are among terms two radically different kinds, whose difference 
constitutes the truth underlying the doctrine of substance and attribute. 
There are terms which can never occur except as terms ; such are points, 
instants, (colours, sounds, bits of matter, and generally terms of the kind 
of which existents consist. There are, on the other hand, terms which 
can occur otherwise than as terms; such are being, adjectives generally, 
and relations. Such terms we agreed to call concepts *. It is the presence 
of concepts not occurring as terms which distinguishes propositions from 
mere concepts ; in every proposition there is at least one more concept 
than there are terms. The traditional view — which may be called the 
subject -predicate theory — holds that in every proposition there is one 
term, the subject, and one concept which is not a term, the predicate. 
This view, for many reasons, must be abandoned f. The smallest 
departure from the traditional opinion lies in holding that, where 
propositions are not reducible to the subject-predicate form, there are 
always two terms only, and one concept which is not a term. (The 
two terms may, of course, be complex, and may each contain concepts 
which are not terms.) This gives the opinion that relations are always 
between only two terms ; for a relation may be defined as any concept 
which occurs in a proposition containing more than one term. But 
there seems no a priori reason for limiting relations to two terms, 
and there are instances which lead to an opposite view. In the first 
place, when the concept of a number is asserted of a collection, if the 
collection has n terms, there are 7i terms, and only one concept (namely 
n) which is not a term. In the second place, such relations as those 
of an existent to the place and time of its existence are only reducible 
by a very cumbrous method to relations of two terms;]:. If, however, 
the reduction be held essential, it seems to be always formally possible, 

"^ See Part I^ Chap. iv. 

t See The Philosophy of Leibniz , by the present author_, Cambridge^ 1900 ; 
Chapter ii^ § 10. 

X See Part VII, Chap. liv. 



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200-202] The vieaning of Order 213 

by compounding part of the proposition into one complex term, and 
then asserting a relation between this part and the remainder, which 
can be similarly reduced to one term. There may be cases where this 
is not possible, but I do not know of them. The question whether such 
a formal reduction is to be always undertaken is not, however, so far 
as I have been able to discover, one of any great practical or theoretical 
importance. 

201. There is thus no valid a priori reason in favour of analyzing 
between into a relation of two relations, if a triangular relation seems 
otherwise preferable. The other reason in favour of the analysis of 
between is more considerable. So long as between is a triangular relation 
of the terms, it must be taken either as indefinable, or as involving a 
reference to some transitive asymmetrical relation. But if we make 
between consist essentially in the opposition of two relations belonging 
to one term, there seems to be no longer any undue indeterminateness. 
Against this view we may urge, however, that no reason now appears 
why the relations in question should have to be transitive, and that — 
what is more important — the very meaning of betxveen involves the 
terms, for it is they, and not their relations, that have order. And 
if it were only the relations that were relevant, it would not be necessary, 
as in fact it is, to particularize them by the mention of the terms 
between which they hold. Thus on the whole, the opinion that between 
is not a triangular relation must be abandoned. 

202. (3) We come now to the view that between is an ultimate 
and indefinable relation. In favour of this view it might be urged that, 
in all our ways of generating open series, we could see that cases of 
between did arise, and that we could apply a test to suggested definitions. 
This seems to show that the suggested definitions were merely conditions 
which imply relations of between^ and were not true definitions of this 
relation. The question : Do such and such conditions insure that y 
shall be between x and z "^ is always one which we can answer, without 
having to appeal (at least consciously) to any previous definition. And 
the unanalyzable nature of between may be supported by the fact that 
the relation is symmetrical with respect to the two extremes, which was 
not the case with the relations of pairs from which between was inferred. 
There is, however, a very grave difficulty in the way of such a view, and 
that is, that sets of terms have many different orders, so that in one we 
may have y between x and <s, while in another we have x between 
y and ^*. This seems to show that between essentially involves reference 
to the relations from which it is inferred. If not, we shall at least have 
to admit that these relations are relevant to the genesis of series ; for 
series require imperatively that there should be at most one relevant 

■^ This case is illustrated by the rational numbers^ which may be taken in order 
of magnitude, or in one of the orders {e.g. the logical order) in which they are 
denumerable. The logical order is the order 1, 2, 1/2, 3, 1/3, 2/3, 4, 



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214 Order [ohap. xxv 

relation of between among three terms. Hence we must, apparently, 
allow that between is not the sole source of series, but must always be 
supplemented by the mention of some transitive asymmetrical relation 
with respect to which the betweenness arises. The most that can be 
said is, that this transitive asymmetrical relation of two terms may 
itself be logically subsequent to, and derived from, some relation of 
three terms, such as those considered in Chapter xxiv, in the fourth way 
of generating series. When such relations fulfil the axioms which were 
then mentioned, they lead of themselves to relations between pairs of 
terms. For we may say that b precedes c when acd implies bcd^ and 
that b follows c when abd implies cbd, where a and d are fixed terms. 
Though such relations are merely derivative, it is in virtue of them 
that between occurs in such cases. Hence we seem finally compelled to 
leave the reference to an asymmetrical relation in our definition. We 
shall therefore say : 

A term y is between two terms x and z with reference to a transitive 
asymmetrical relation R when xRy and yRz. In no other case can y 
be said properly to be between x and z ; and this definition gives not 
merely a criterion, but the very meaning of betweenness. 

203. We have next to consider the meaning of separation of 
couples. This is a more complicated relation than betioeen^ and was 
but little considered until elliptic Geometry brought it into prominence. 
It has been shown by Vailati* that this relation, like between^ always 
involves a transitive asymmetrical relation of two terms ; but this rela- 
tion of a pair of terms is itself relative to three other fixed terms of the 
set, as, in the case of between^ it was relative to two fixed terms. It is 
further sufficiently evident that wherever there is a transitive asymme- 
trical relation, which relates every pair of terms in a collection of not 
less than four terms, there there are pairs of couples having the relation 
of separation. Thus we shall find it possible to express separation, as 
well as betxveen^ by means of transitive asymmetrical relations and their 
terms. But let us first examine directly the meaning of separation. 

We may denote the fact that a and c are separated by b and d by 
the symbol abed. If, then, a, &, c, d^ e be any five terms of the set we 
require the following properties to hold of the relation of separation (of 
which, it will be observed, only the last involves five 
terms) : 

1. abed — bade. ^J. \j 
%. abed = adeb. 

2. abed excludes acbd, 

4. We must have abed or acdb or adbe. 

5. abed and acde together imply abdej", 

^ Rivista di Matematica,Y, pp. 75 — 78. See also Pieri^ IFrincipii della Geometria 
di Posizione, Turin^ 1898^ § 7. 

t These five properties are taken from Vailati;, loc. cit. and ib. p. 183. 




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202-204] The meaning of Order 215 

These properties may be illustrated by the consideration of five 
points on a circle, as in the accompanying figure. Whatever relation 
of two pairs of terms possesses these properties we shall call a relation 
of separation between the pairs. It will be seen that the relation is 
symmetrical, but not in general transitive. 

204. Wherever we have a transitive asymmetrical relation R be- 
tween any two terms of a set of not less than four terms, the relation of 
separation necessarily arises. For in any series, if four terms have the 
order abcd^ then a and c are separated by h and d ; and every transitive 
asymmetrical relation, as we have seen, provided there are at least two 
consecutive instances of it, gives rise to a series. Thus in this case, 
separation is a mere extension of hetween : if R be asymmetrical and 
transitive, and aRh^ hRc^ cRd^ then a and c are separated by h and d. 
The existence of such a relation is therefore a sufficient condition of 
separation. 

It is also a necessary condition. For, suppose a relation of separation 
to exist, and let a^ 6, c, d^ e be five terms of the set to which the relation 
applies. Then, considering <2, 5, c as fixed, and d and e as variable, 
twelve cases may arise. In virtue of the five fundamental properties, we 
may introduce the symbol ahcde to denote that, striking out any one 
of these five letters, the remaining four have the relation of separation 
which is indicated by the resulting symbol. Thus by the fifth property, 
abed and acde imply ahcde^. Thus the twelve cases arise from permuting 
d and ^, while keeping <2, &, c fixed. (It should be observed that it 
makes no difference whether a letter appears at the end or the beginning: 
i.e. ahede is the same case as eahed. We may therefore decide not to put 
either d or e before a.) Of these twelve cases, six will have d before e^ 
and six will have e before d. In the first six cases, we say that, with 
respect to the sense abe^ d precedes e ; in the other six cases, we say that 
e precedes d. In order to deal with limiting cases, we shall say further 
that a precedes every other term, and that b precedes cf. We shall then 
find that the relation of preceding is asymmetrical and transitive, and 
that every pair of terms of our set, is such that one precedes and the 
other follows. In this way our relation of separation is reduced, formally 
at least, to the combination of " a precedes 6,^' " b precedes 6',"' and " c 
precedes <i." 

The above reduction is for many reasons highly interesting. In the 
first place, it shows the distinction between open and closed series to be 
somewhat superficial. For although our series may initially be of the 
sort which is called closed, it becomes, by the introduction of the above 
transitive relation, an open series, having a for its beginning, but having 



"^ The argument is somewhat tedious^ and I therefore omit it. It will be found 
in Vailati^ loc. cit. 

t Pieri^ op. cit. p. 32. 



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216 Order [chap, xxv 

possibly no last term, and not in any sense returning to a. Again it is 
of the highest importance in Geometry, since it shows how order may 
arise on the elliptic straight line, by purely projective considerations, 
in a manner which is far more satisfactory than that obtained from 
von Staudt's construction*. And finally, it is of great importance as 
unifying the two sources of order, between and separation ; since it 
shows that transitive asymmetrical relations are always present where 
either occurs, and that either implies the other. For, by the relation of 
preceding, we can say that one term is between two others, although we 
started solely from separation of pairs. 

205. At the same time, the above reduction (and also, it would 
seem, the corresponding reduction in the case of between) cannot be 
allowed to be more than formal. That is, the three terms <x, &, c by 
relation to which our transitive asymmetrical relation was defined, are 
essential to the definition, and cannot be omitted. The reduction shows 
no reason for supposing that there is any transitive asymmetrical relation 
independent of all other terms than those related, though it is arbitrary 
what other terms we choose. And the fact that the term a, which is 
not essentially peculiar, appears as the beginning of the series, illustrates 
this fact. Where there are transitive asymmetrical relations independent 
of all outside reference, our series cannot have an arbitrary beginning, 
though it may have none at all. Thus the four-term relation of sepa- 
ration remains logically prior to the resulting two-term relation, and 
cannot be analyzed into the latter. 

206. But when we have said that the reduction is formal, we have 
not said that it is irrelevant to the genesis of order. On the contrary, 
it is just because such a reduction is possible that the four-term relation 
leads to order. The resulting asymmetrical transitive relation is in 
reality a relation of five terms ; but when three of these are kept fixed, 
it becomes asymmetrical and transitive as regards the other two. Thus 
although between applies to such series, and although the essence of 
order consists, here as elsewhere, in the fact that one term has, to two 
others, converse relations which are asymmetrical and transitive, yet 
such an order can only arise in a collection containing at least fiye terms, 
because ^\e terms are needed for the characteristic relation. And it 
should be observed that all series, when thus explained, are open series, 
in the sense that there is some relation between pairs of terms, no power 
of which is equal to its converse, or to identity. 

207. Thus finally, to sum up this long and complicated discussion : 
The six methods of generating series enumerated in Chapter xxiv are all 
genuinely distinct ; but the second is the only one which is fundamental, 

"^ The advantages of this method are evident from Pieri's work quoted above^ 
where many things which seemed incapable of projective proof are rigidly deduced 
from projective premisses. See Part VI^ Chap. xlv. 



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204-207] The meaning of Order 217 

and the other ^yq agree in this, that they are all reducible to the second. 
Moreover, it is solely in virtue of their reducibility to the second that 
they give rise to order. The minimum ordinal proposition, which can 
always be made wherever there is an order at all, is of the form : " y is 
between x and z'''' ; and this proposition means ; " There is some 
asymmetrical transitive relation which holds between x and y and 
between y and ^.'^ This very simple conclusion might have been guessed 
from the beginning; but it was only by discussing all the apparently 
exceptional cases that the conclusion could be solidly estabhshed. 



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CHAPTER XXVI. 



ASYMMETRICAL RELATIONS. 

208. We have now seen that all order depends upon transitive 
asymmetrical relations. As such relations are of a kind which traditional 
logic is unwilling to admit, and as the refusal to admit them is one of 
the main sources of the contradictions which the Critical Philosophy has 
found in mathematics, it will be desirable, before proceeding further, to 
make an excursion into pure logic, and to set forth the grounds which 
make the admission of such relations necessary. At a later stage (in 
Part VI, Chap, li), I shall endeavour to answer the general objections 
of philosophers to relations ; for the present, I am concerned only with 
asymmetrical relations. 

Relations may be divided into four classes, according as they do 
or do not possess either of two attributes, transitiveness * and symmetry. 
Relations such that xRy always implies yRx are called symmetrical ; 
relations such that xRy^ yRz together always imply ocRz are called 
transitive. Relations which do not possess the first property I shall 
call not symmetrical:, relations which do possess the opposite property, 
i.e. for which xRy always excludes yRsc^ I shall call asymmetrical. 
Relations which do not possess the second property I shall call not 
transitive ; those which possess the property that ocRy^ yRz always 
exclude xRz I shall call intransitive. AH these cases may be illus- 
trated from human relationships. The relation brother or sister is 
symmetrical, and is transitive if we allow that a man may be his 
own brother, and a woman her own sister. The relation brother is not 
symmetrical, but is transitive. Half-brother or half-sister is symmetrical 
but not transitive. Spouse is symmetrical but intransitive ; descendant 
is asymmetrical but transitive. Half-brother is not symmetrical and not 
transitive ; if third marriages were forbidden, it would be intransitive. 
Son-in-law is asymmetrical and not transitive ; if second marriages were 
forbidden, it would be intransitive. Brother-in-law is not symmetrical 

■^ This term appears to have been first used in the present sense by De Morgan ; 
see Camh. Phil. Trans, ix, p. 104 ; x_, p. 346. The term is now in general use. 



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208-210] Asymmetrical Relations 219 

and not transitive. Finally, jTa^^^r is both asymmetrical and intransitive. 
Of not-transitive but not intransitive relations there is, so far as I know, 
only one important instance, namely diversity ; of not-symmetrical but not 
asymmetrical relations there seems to be similarly only one important 
instance, namely implication. In other cases, of the kind that usually 
occur, relations are either transitive or intransitive, and either sym- 
metrical or asymmetrical. 

209. Relations which are both symmetrical and transitive are formally 
of the nature of equality. Any term of the field of such a relation has the 
relation in question to itself, though it may not have the relation to any 
other term. For denoting the relation by the sign of equality, if a be 
of the field of the relation, there is some term h such that a= b. If 
a and b be identical, then a — a. But if not, then, since the relation 
is symmetrical, b = a; since it is transitive, and we have a = &, b = a, 
it follows that a = a. The property of a relation which insures that 
it holds between a term and itself is called by Peano reflexiveness, and 
he has shown, contrary to what was previously believed, that this 
property cannot be inferred from symmetry and transitiveness. For 
neither of these properties asserts that there is a 5 such that a — b^ but 
only what follows in case there is such a b ; and if there is no such 6, 
then the proof of a = a fails*. This property of reflexiveness, however, 
introduces some difficulty. There is only one relation of which it is true 
without limitation, and that is identity. In all other cases, it holds 
only of the terms of a certain class. Quantitative equality, for example, 
is only reflexive as applied to quantities ; of other terms, it is absurd 
to assert that they have quantitative equality with themselves. Logical 
equality, again, is only reflexive for classes, or propositions, or relations. 
Simultaneity is only reflexive for events, and so on. Thus, with any 
given symmetrical transitive relation, other than identity, we can only 
assert reflexiveness within a certain class ; and of this class, apart from 
the principle of abstraction (already mentioned in Part III, Chap, xix, 
and shortly to be discussed at length), there need be no definition 
except as the extension of the transitive symmetrical relation in question. 
And when the class is so defined, reflexiveness within that class, as we 
have seen, follows from transitiveness and symmetry. 

210. By introducing what I have called the principle of abstraction f, 
a somewhat better account of reflexiveness becomes possible. Peano has 
defined J a process which he calls definition by abstraction, of which, as 
he shows, frequent use is made in Mathematics. This process is as 

"^ See e.g. Revue de Mathematiques , T. vii^ p. 22 ; Notations de Logique Mathe- 
matique, Turin, 1894, p. 45, F. 1901, p. 193. 

t An axiom virtually identical with this principle, but not stated with the 
necessary precision, and not demonstrated, will be found in De Morgan, Gamh. PhiL 
Trans. Vol. x, p. 345. 

I Notations de Logique Mathematique, p. 45. 



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220 Orde7^ [chap, xxvi 

follows : when there is any relation which is transitive, symmetrical and 
(within its field) reflexive, then, if this relation holds between u and v^ 
we define a new entity ^ {it)^ which is to be identical with (^ {v). Thus 
our relation is analyzed into sameness of relation to the new term 
(/) {u) or (j6 {v). Now the legitimacy of this process, as set forth by 
Peano, requires an axiom, namely the axiom that, if there is any 
instance of the relation in question, then there is such an entity as 
(/) {u) or (/) {v). This axiom is my principle of abstraction, which, 
precisely stated, is as follows : " Every transitive symmetrical relation, 
of which there is at least one instance, is analyzable into joint possession 
of a new relation to a new term, the new relation being such that no 
term can have this relation to more than one term, but that its converse 
does not have this property." This principle amounts, in common 
language, to the assertion that transitive symmetrical relations arise 
from a common property, with the addition that this property stands, 
to the terms which have it, in a relation in which nothing else stands 
to those terms. It gives the precise statement of the principle, often 
applied by philosophers, that symmetrical transitive relations always 
spring from identity of content. Identity of content is, however, an 
extremely vague phrase, to which the above proposition gives, in the 
present case, a precise signification, but one which in no way answers 
the purpose of the phrase, which is, apparently, the reduction of relations 
to adjectives of the related terms. 

It is now possible to give a clearer account of the reflexive property. 
Let R be our symmetrical relation, and let S be the asymmetrical 
relation which two terms having the relation R must have to some 
third term. Then the proposition ocRy is equivalent to this : " There 
is some term a such that xSa and ySa^ Hence it follows that, if oc 
belongs to what we have called the domain of S^ i.e. if there is any 
term a such that a^Sa, then xRx ; for ojRaj is merely a^Sa and xSa. It 
does not of course follow that there is any other term y such that ^Ry^ 
and thus Peano's objections to the usual proof of reflexiveness are valid. 
But by means of the analysis of symmetrical transitive relations, we 
obtain the proof of the reflexive property, together with the exact 
limitation to which it is subject. 

211. We can now see the reason for excluding from our accounts 
of the methods of generating series a seventh method, which some 
readers may have expected to find. This is the method in which 
position is merely relative — a method which, in Chap, xix, § 154, 
we rejected as regards quantity. As the whole philosophy of space 
and time is bound up with the question as to the legitimacy of this 
method, which is in fact the question as to absolute and relative 
position, it may be well to give an account of it here, and to show 
how the principle of abstraction leads to the absolute theory of position. 

If we consider such a series as that of events, and if we refuse to 



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210-212] Asymmetrical Relations 221 

allow absolute time, we shall have to admit three fandamental relations 
among events, namely, simultaneity, priority, and posteriority. Such a 
theory may be formally stated as follows : Let there be a class of terms, 
such that any two, oc and «/, have either an asymmetrical transitive 
relation P, or the converse relation P, or a symmetrical transitive 
relation R. Also let xRy^ yPz imply xPz^ and let ocPy^ yRz imply ocPz. 
Then all the terms can be arranged in a series, in which, however, there 
may be many terms which have the same place in the series. This 
place, according to the relational theory of position, is nothing but 
the transitive symmetrical relation i? to a number of other terms. But 
it follows from the principle of abstraction that there is some relation S^ 
such that, if xRy^ there is some one entity t for which xSt^ ySt. We 
shall then find that the different entities ^, corresponding to different 
groups of our original terms, also form a series, but one in which 
any two different terms have an asymmetrical relation (formally, the 
product SRS). These terms t will then be the absolute positions of 
our ^'"s and y\s^ and our suppx3sed seventh method of generating series 
is reduced to the fundamental second method. Thus there will be no 
series having only relative position, but in all series it is the positions 
themselves that constitute the series*. 

212. We are now in a position to meet the philosophic dislike of 
relations. The whole account of order given above, and the present 
argument concerning abstraction, will be necessarily objected to by 
those philosophers — and they are, I fear, the major part — who hold 
that no relations can possess absolute and metaphysical validity. It 
is not my intention here to enter upon the general question, but merely 
to exhibit the objections to any analysis of asymmetrical relations. 

It is a common opinion — often held unconsciously, and employed 
in argument, even by those who do not explicitly advocate it — that 
all propositions, ultimately, consist of a subject and a predicate. When 
this opinion is confronted by a relational proposition, it has two ways 
of dealing with it, of which the one may be called monadistic, 
the other monistic. Given, say, the proposition aRh^ where R is some 
relation, the monadistic view will analyse this into two propositions, 
which we may call ar^ and ^rg, which give to <2 and h respectively 
adjectives supposed to be together equivalent to R. The monistic 
view, on the contrary, regards the relation as a property of the whole 
composed of a and h. and as thus equivalent to a proposition which 
we may denote by {ab)r. Of these views, the first is represented by 
Leibniz and (on the whole) by Lotze, the second by Spinoza and 
Mr Bradley. Let us examine these views successively, as applied to 



^ A formal treatment of relative position is given by Schroder^ Su7' une eoctension 
d% Videe d'ordre, Coiigres, Vol. iii^ p. 235. 



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222 Order [chap, xxvi 

asymmetrical relations ; and for the sake of definiteness, let us take 
the relations of greater and less. 

213. The monadistic view is stated with admirable lucidity by 
Leibniz in the following passage* : 

" The ratio or proportion between two lines L and M may be 
conceived three several ways ; as a ratio of the greater L to the 
lesser il/; as a ratio of the lesser M to the greater Z; and lastly, as 
something abstracted from both, that is, as the ratio between L and J/, 
without considering w^hich is the antecedent, or which the consequent ; 
which the subject, and which the object.... In the first way of considering 
them, L the greater, in the second M the lesser, is the subject of that 
accident which philosophers call relation. But which of them will be 
the subject, in the third way of considering them ? It cannot be said 
that both of them, L and M together, are the subject of such an 
accident ; for if so, we should have an accident in two subjects, with 
one leg in one, and the other in the other ; which is contrary to the 
notion of accidents. Therefore we must say that this relation, in this 
third way of considering it, is indeed out of the subjects ; but being 
neither a substance nor an accident, it must be a mere ideal thing, 
the consideration of which is nevertheless useful." 

214. The third of the above ways of considering the relation of 
greater and less is, roughly speaking, that which the monists advocate, 
holding, as they do, that the whole composed of L and M is one subject, 
so that their way of considering ratio does not compel us, as Leibniz 
supposed, to place it among bipeds. For the present our concern is only 
with the first two ways. In the first way of considering the matter, we 
have "L is (greater than i)/)," the words in brackets being considered 
as an adjective of L. But when we examine this adjective it is at once 
evident that it is complex: it consists, at least, of the parts greater 
and ilf, and both these parts are essential. To say that L is greater 
does not at all convey our meaning, and it is highly probable that M is 
also greater. The supposed adjective of L involves some reference to M \ 
but what can be meant by a reference the theory leaves unintelligible. 
An adjective involving a reference to M is plainly an adjective which is 
relative to M^ and this is merely a cumbrous way of describing a relation. 
Or, to put the matter otherwise, if L has an adjective corresponding 
to the fact that it is greater than i/, this adjective is logically sub- 
sequent to, and is merely derived from, the direct relation of L to M. 
Apart from M^ nothing appears in the analysis of L to differentiate it 
from M ; and yet, on the theory of relations in question, L should differ 
intrinsically from M. Thus we should be forced, in all cases of asym- 
metrical relations, to admit a specific difference between the related 
terms, although no analysis of either singly will reveal any relevant 

^ Phil. Werke, Gerhardt's ed.^ Vol. vii^ p. 401. 



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212-214] Asymmetrical Relations 223 

property which it possesses and the other lacks. For the monadistic 
theory of relations, this constitutes a contradiction ; and it is a contra- 
diction which condemns the theory from which it springs*. 

Let us examine further the application of the monadistic theory to 
quantitative relations. The proposition " A is greater than B " is to be 
analyzable into two propositions, one giving an adjective to A^ the 
other giving one to B. The advocate of the opinion in question will 
probably hold that A and B are quantities, not magnitudes, and will 
say that the adjectives required are the magnitudes of A and B. But 
then he will have to admit a relation between the magnitudes, which 
will be as asymmetrical as the relation which the magnitudes were to 
explain. Hence the magnitudes will need new adjectives, and so on 
ad iriflnitum ; and the infinite process will have to be completed before 
any meaning can be assigned to our original proposition. This kind 
of infinite process is undoubtedly objectionable, since its sole object 
is to explain the meaning of a certain proposition, and yet none of its 
steps bring it any nearer to that meaning f. Thus we cannot take 
the magnitudes of A and B as the required adjectives. But further, 
if we take any adjectives whatever except such as have each a reference 
to the other term, we shall not be able, even formally, to give any 
account of the relation, without assuming just such a relation between 
the adjectives. For the mere fact that the adjectives are different will 
yield only a symmetrical relation. Thus if our two terms have different 
colours we find that A has to B the relation of differing in colour, 
a relation which no amount of careful handling will render asymmetrical. 
Or if we were to recur to magnitudes, we could merely say that A and 
B differ in magnitude, which gives us no indication as to which is 
the greater. Thus the adjectives of A and B must be, as in Leibniz's 
analysis, adjectives .having a reference each to the other term. The 
adjective of A must be "greater than J5,'' and that of B must be "less 
than ^." Thus A and B differ, since they have different adjectives — 
B is not greater than J5, and A is not less than A — but the adjectives 
are extrinsic, in the sense that ^'s adjective has reference to i?, and 
B's to A. Hence the attempted analysis of the relation fails, and we 



■^ See a paper on '^'^The Relations of Number and Quantity/' Miiid, N.S. No. 23. 
This paper was written while I still adhered to the monadistic theory of relations : 
the contradiction in question^ therefore^ was regarded as inevitable. The following 
passage from Kant raises the same point : '^'^Die rechte Hand ist der linken ahnlich 
und gleich^ und wenn man bios auf eine derselben allein sieht^ auf die Proportion 
der Lage der Theile unter einander und auf die Grosse des Ganzen^ so muss eine 
vollstandige Beschreibung der einen in alien Stucken auch von der andern gelten." 
(Von dem ersten Grunde des UnterscMedes der Gegenden im Raume, ed. Hart. Vol. ii, 
p. 889.) 

t Where an infinite process of this kind is required we are necessarily dealing 
with a proposition which is an infinite unity^ in the sense of Part 11^ Chap. xvii. 



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224 Order [chap, xxvi 

are forced to admit what the theory was designed to avoid, a so-called 
" external '' relation, i.e. one implying no complexity in either of the 
related terms. 

The same result may be proved of asymmetrical relations generally, 
since it depends solely upon the fact that both identity and diversity 
are symmetrical. Let a and h have an asymmetrical relation i?, so 
that aRh and bRa. Let the supposed adjectives (which, as we have 
seen, must each have a reference to the other term) be denoted by /5 
and a respectively. Thus our terms become a/S and ha. a involves 
a reference to a, and /3 to b ; and a and /S differ, since the relation 
is asymmetrical. But a and b have no intrinsic differences corresponding 
to the relation iZ, and prior to it ; or, if they have, the points of 
difference must themselves have a relation analogous to jR, so that 
nothing is gained. Either a or /3 expresses a difference between a 
and b, but one which, since either a or /S involves reference to a term 
other than that whose adjective it is, so far from being prior to J?, 
is in fact the relation R itself. And since a and /3 both presuppose J?, 
the difference between a and /3 cannot be used to supply an intrinsic 
difference between a and b. Thus we have again a difference without 
a prior point of difference. This shows that some asymmetrical rela- 
tions must be ultimate, and that at least one such ultimate asymmetrical 
relation must be a component in any asymmetrical relation^ tjiat may be 
suggested. 

It is easy to criticize the monadistic theory from a general stand- 
point, by developing the contradictions which spring from the relations 
of the terms to the adjectives into which our first relation has been 
analyzed. These considerations, which have no special connection with 
asymmetry, belong to general philosophy, and have been urged by 
advocates of the monistic theory. Thus Mr Bradley says of the mona- 
distic theory * : " We, in brief, are led by a principle of fission which 
conducts us to no end. Every quality in relation has, in consequence, 
a diversity within its own nature, and this diversity cannot immediately 
be asserted of the quality. Hence the quality must exchange its unity 
for an internal relation. But, thus set free, the diverse aspects, because 
each something in relation, must each be something also beyond. This 
diversity is fatal to the internal unity of each ; and it demands a new 
relation, and so on without limit." It remains to be seen whether the 
monistic theory, in avoiding this difficulty, does not become subject to 
others quite as serious. 

215. The monistic theory holds that every relational proposition 
aRb is to be resolved into a proposition concerning the whole which 
a and b compose — a proposition which we may denote by {ab)r. This 
view, like the other, may be examined with special reference to asym- 

■^ Appearance and Reality, 1st edition^ p. 31. 



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214, 215] Asyimnetrical Relations 225 

metrical relations, or from the standpoint of general philosophy. We 
are told, by those who advocate this opinion, that the whole contains 
diversity within itself, that it synthesizes differences, and that it performs 
other similar feats. For my part, I am unable to attach any precise 
significance to these phrases. But let us do our best. 

The proposition " a \^ greater than 6,'" we are told, does not really 
say anything about either a or 6, but about the two together. Denoting 
the Avhole which they compose by {ah\ it says, we will suppose, '' (ah) 
contains .diversity of magnitude." Now to this statement — neglecting 
for the present all general arguments — there is a special objection in 
the case of asymmetry, {ah) is symmetrical with regard to a and h^ 
and thus the property of the whole will be exactly the same in the case 
where a is greater than h as in the case where h is greater than a. 
Leibniz, who did not accept the monistic theory, and had therefore 
no reason to render it plausible, clearly perceived this fact, as appears 
from the above quotation. For, in his third way of regarding ratio, 
we do not consider which is the antecedent, which the consequent; 
and it is indeed sufficiently evident that, in the whole {ah) as such, 
there is neither antecedent nor consequent. In order to distinguish 
a whole {ah) from a whole {ha\ as we must do if we are to explain 
asymmetry, we shall be forced back from the whole to the parts and 
their relation. For {ah) and {ha) consist of precisely the same parts, 
and differ in no respect whatever save the sense of the relation between 
a and h. " a is greater than &"" and '' 6 is greater than a'''' are proposi- 
tions containing precisely the same constituents, and giving rise therefore 
to precisely the same whole ; their difference lies solely in the fact that 
greater is, in the first case, a relation of a to ^, in the second, a relation 
of h to a. Thus the distinction of sense, i.e. the distinction between an 
asymmetrical relation and its converse, is one Avhich the monistic theory 
of relations is wholly unable to explain. 

Arguments of a more general nature might be multiplied almost 
indefinitely, but the foUoAving argument seems peculiarly relevant. The 
relation of whole and part is itself an asymmetrical relation, and the 
whole — as monists are peculiarly fond of telling us — is distinct from all 
its parts, both severally and collectively. Hence when we say "a is 
part of 6," we really mean, if the monistic theory be correct, to assert 
something of the whole composed of a and Z>, which is not to be 
confounded with h. If the proposition concerning this new whole be not 
one of whole and part there will be no true judgments of whole and 
part, and it will therefore be false to say that a relation between the 
parts is really an adjective of the whole. If the new proposition is one 
of whole and part, it will require a new one for its meaning, and so on. 
If, as a desperate measure, the monist asserts that the whole composed 
of a and h is not distinct from Z>, he is compelled to admit that a whole 
is the sum (in the sense of Symbolic Logic) of its parts, which, besides 

R. 15 



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226 Order [chap, xxvi 

being an abandonment of bis whole position, renders it inevitable that 
the whole should be symmetrical as regards its parts — a view which we 
have already seen to be fatal. And hence we find monists driven to 
the view that the only true whole, the Absolute, has no parts at all, 
and that no propositions in regard to it or anything else are quite 
true — a view which, in the mere statement, unavoidably contradicts 
itself. And surely an opinion which holds all propositions to be in the 
end self-contradictory is sufficiently condemned by the faict that, if it 
be accepted, it also must be self-contradictory. 

216. We have now seen that asymmetrical relations are unin- 
telligible on both the usual theories of relation*. Hence, since such 
relations are involved in Number, Quantity, Order, Space, Time, and 
Motion, we can hardly hope for a satisfactory philosophy of Mathematics 
so long as we adhere to the view that no relation can be " purely 
external." As soon, however, as we adopt a different theory, the logical 
puzzles, which have hitherto obstructed philosophers, are seen to be 
artificial. Among the terms commonly regarded as relational, those 
that are symmetrical and transitive — such as equality and simultaneity — 
are capable of reduction to what has been vaguely called identity of 
content, but this in turn must be analyzed into sameness of relation 
to some other term. For the so-called properties of a term are, in fact, 
only other terms to which it stands in some relation ; and a common 
property of two terms is a term to which both stand in the same 
relation. 

The present long digression into the realm of logic is necessitated 
by the fundamental importance of order, and by the total impossibility 
of explaining order without abandoning the most cherished and wide- 
spread of philosophic dogmas. Everything depends, where order is 
concerned, upon asymmetry and difference of sense, but these two concepts 
are unintelligible to the traditional logic. In the next chapter we shall 
have to examine the connection of difference of sense with what appears 
in Mathematics as difference of sign. In this examination, though some 
pure logic will still be requisite, we shall approach again to mathematical 
topics ; and these will occupy us wholly throughout the succeeding chapters 
of this Part. 

* The grounds of these theories will be examined from a more general point of 
view in Part \l, Chap. li. 



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CHAPTER XXVII. 



DIFFERENCE OF SENSE AND DIFFERENCE OF SIGN. 

217. We have now seen that order depends upon asymmetrical 
relations, and that these always have two senses, as before and after, 
greater and less, east and west, etc. The difference of sense is closely 
connected (though not identical) with the mathematical difference of 
sign. It is a notion of fundamental importance in Mathematics, and 
is, so far as I can see, not explicable in terms of any other notions. 
The first philosopher who realized its importance would seem to be Kant. 
In the Versuch den Begiiff der negativen Grosse in die Weltweisheit 
einzufuhren (1763), we find him aware of the difference between logical 
opposition and the opposition of positive and negative. In the discussion 
Von dem erst en Grunde des UnterscMedes der Gegenden irn Raume (1768), 
we find a full realization of the importance of asymmetry in spatial 
• relations, and a proof, based on this fact, that space cannot be wholly 
relational*. But it seems doubtful whether he realized the connection of 
this asymmetry with difference of sign. In 1763 he certainly was not 
aware of the connection, since he regarded pain as a negative amount of 
pleasure, and supposed that a great pleasure and a small pain can be 
added to give a less pleasure f — a view which seems both logically and 
psychologically false. In the Prolegoviena (§ 13), as is well known, 
he made the asymmetry of spatial relations a ground for regarding space 
as a mere form of intuition, perceiving, as appears from the discussion 
of 1768, that space could not consist, as Leibniz supposed, of mere 
relations among objects, and being unable, owing to his adherence to 
the logical objection to relations discussed in the preceding chapter, 
to free from contradiction the notion of absolute space with asym- 
metrical relations between its points. Although I cannot regard this 
later and more distinctively Kantian theory as an advance upon that 
of 1768, yet credit is undoubtedly due to Kant for having first called 
attention to the logical importance of asymmetrical relations. 

* See especially ed. Hart, Vol. II, pp. 386, 391. 
t Ed. Hart, Vol. II, p. 83. 

15—2 



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228 Order [chap, xxvii 

218. By difference of sense I mean, in the present discussion at least, 
the difference between an asymmetrical relation and its converse. It is a 
fundamental logical fact that, given any relation J?, and any two terms 
a, Z>, there are two propositions to be formed of these elements, the one 
relating a to h (which I call aRh)^ the other {hRa) relating h to a. These 
two propositions are always different, though sometimes (as in the case 
of diversity) either implies the other. In other cases, such as logical im- 
plication, the one does not imply either the other or its negation; while 
in a third set of cases, the one implies the negation of the other. It is 
only in cases of the third kind that I shall speak of difference of sense. 
In these cases, aRh excludes hRa. But here another fundamental logical 
fact becomes relevant. In all cases where aRh does not imply hRa there 
is another relation, related to i?, which must hold between h and a. That 
is, there is a relation R such that aRh implies hRa ; and further, hRa 
implies aRh. The relation of R to R is difference of sense. This 
relation is one-one, symmetrical, and intransitive. Its existence is the 
source of series, of the distinction of signs, and indeed of the greater 
part of mathematics. 

219. A question of considerable importance to logic, and especially 
to the theory of inference, may be raised with regard to difference of sense. 
Are aRh and hRa really different propositions, or do they only differ 
linguistically ? It may be held that there is only one relation R^ and 
that all necessary distinctions can be obtained from that between aRh 
and hRa. It may be said that, owing to the exigencies of speech and 
writing, we are compelled to mention either a or h first, and that this gives 
a seeming difference between " a is greater than h " and " h is less than 
^"; but that, in reality, these two propositions are identical. But if 
we take this view we shall ffnd it hard to explain the indubitable 
distinction between greater and less. These two words have certainly 
each a meaning, even when no terms are mentioned as related by them. 
And they certainly have different meanings, and are certainly relations. 
Hence if we are to hold that " a is greater than h '' and " h is less than a '^ 
are the same proposition, we shall have to maintain that both greater 
and less enter into each of these propositions, which seems obviously 
false ; or else w^e shall have to hold that what really occurs is neither 
of the two, but that third abstract relation mentioned by Leibniz in the 
passage quoted above. In this case the difference between greater and 
less would be one essentially involving a reference to the terms a and h. 
But this view cannot be maintained without circularity ; for neither the 
greater nor the less is inherently the antecedent, and we can only say 
that, when the greater is the antecedent, the relation is greater ; when 
the less, the relation is less. Hence, it would seem, we must admit that 
R and R are distinct relations. We cannot escape this conclusion by 
the analysis into adjectives attempted in the last chapter. We there 



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21|-221] Difference of Sense and Dfference of Sign 229 

analyzed aRh into aj3 and ha. But, corresponding to every h^ there will 

be two adjectives, /3 and /^, and corresponding to every a there will also 
be two, a and a. Thus if R be gi^eater^ a will be " greater than A '''' 
and a " less than A,''"' or vice versa. But the difference between a and a 
presupposes that between greater and less, between R and i?, and therefore 
cannot explain it. Hence R and R must be distinct, and ^^ aRh implies 
hRa '^ must be .a genuine inference. 

I come now to the connection between difference of sense and 
difference of sign. We shall find that the latter is derivative from the 
former, being a difference which only exists between terms which either 
are, or are correlated with, asymmetrical relations. But in certain cases 
we shall find some complications of detail which will demand discussion. 

The difference of signs belongs, traditionally, only to numbers and 
magnitudes, and is intimately associated with addition. It may be 
allowed that the notation cannot be usefully employed where there 
is no addition, and even that, where distinction of sign is possible, 
addition in some sense is in general also possible. But we shall find 
that the difference of sign has no very intimate connection with addition 
and subtraction. To make this clear, we must, in the first place, 
clearly realize that numbers and magnitudes which have no sign are 
radically different from such as are positive. Confusion on this point is 
quite fatal to any just theory of signs. 

220. Taking first finite numbers, the positive and negative numbers 
arise as follows*. Denoting by R the relation between two integers in 
virtue of which the second is next after the first, the proposition mRn 
is equivalent to what is usually expressed by m + 1 = n. But the present 
theory will apply to progressions generally and does not depend upon 
the logical theory of cardinals developed in Part II. In the proposition 
inRn^ the integers m and n are considered, as when they result from the 
logical definition, to be wholly destitute of sign. If now mRn and nRp^ 
we put 7nR^p\ and so on for higher powers. Every power of R is an 
asymmetrical relation, and its converse is easily shown to be the same 

power of R as it is itself of R. Thus viR^^q is equivalent to qR^^m. 
These are the two propositions which are commonly written m + a = q 
and q — a = m. Thus the relations i?% R^ are the true positive and 
negative integers; and these, though associated with a, are both wholly 
distinct from it. Thus in this case the connection with difference of 
sense is obvious and straightforward. 

221. As regards magnitudes, several cases must be distinguished. 
We have (1) magnitudes which are not either relations or stretches, 
(2) stretches, (3) magnitudes which are relations. 

* I give the theory briefly liere^ as it will be dealt with more fully and generally 
in the chapter on Progressions,, § 233. 



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230 Order [char xxvii 

(1) Magnitudes of this class are themselves neither positive nor 
negative. But two such magnitudes, as explained in Part III, determine 
either a distance or a stretch, and these are always positive or negative. 
These are moreover always capable of addition. But since our original 
magnitudes are neither relations nor stretches, the new magnitudes thus 
obtained are of a different kind from the original set. Thus the differ- 
ence of two pleasures, or the collection of pleasures intermediate between 
two pleasures, is not a pleasure, but in the one case a relation, in the 
other a class. 

(2) Magnitudes of divisibility in general have no sign, but when 
they are magnitudes of stretches they acquire sign by correlation. 
A stretch is distinguished from other collections by the fact that it 
consists of all the terms of a series intermediate between two given 
terms. By combining the stretch with one sense of the asymmetrical 
relation which must exist between its end-terms, the stretch itself 
acquires sense, and becomes asymmetrical. That is, we can distinguish 
(1) the collection of terms betiveen a and h without regard to order, (^) 
the terms from a to &, (3) the terms from h to a. Here (2) and (3) are 
complex, being compounded of (1) and one sense of the constitutive 
relation. Of these two, one must be called positive, the other negative. 
Where our series consists of magnitudes, usage and the connection with 
addition have decided that, if a is less than Z>, (2) is positive and (3) is 
negative. But where, as in Geometry, our series is not composed of 
magnitudes, it becomes wholly arbitrary which is to be positive and 
which negative. In either case, we have the same relation to addition, 
which is as follows. Any pair of collections can be added to form a new 
collection, but not any pair of stretches can be added to form a new 
stretch. For this to be possible the end of one stretch must be con- 
secutive to the beginning of the other. In this way, the stretches ah^ he 
can be added to form the stretch ac. If a5, he have the same sense, ae is 
greater than either; if they have different senses, ae is less than one 
of them. In this second case the addition of ah and he is regarded 
as the subtraction of ah and cZ>, he and eh being negative and positive 
respectively. If our stretches are numerically measurable, addition or 
subtraction of their measures will give the measure of the result of 
adding or subtracting the stretches, where these are such as to allow 
addition or subtraction. But the whole opposition of positive and 
negative, as is evident, depends upon the fundamental fact that our 
series is generated by an asymmetrical relation. 

(3) Magnitudes which are relations may be either symmetrical or 
asymmetrical relations. In the former case, if « be a term of the field 
of one of them, the other terms of the various fields, if certain conditions 
are fulfilled*, may be arranged in series according as their relations to a 
are greater or smaller. This arrangement may be different when we choose 

* Cf. § 245. 



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221, 222] Difference of Sense and Dfference of Sign 231 

some term other than a; for the present, therefore, we shall suppose a to 
be chosen once for all. When the terms have been arranged in a series, 
it may happen that some or all places in the series are occupied by more 
than one term; but in any case the assemblage of terms between a and 
some other term m is definite, and leads to a stretch with two senses. 
We may then combine the magnitude of the relation of a to m with one 
or other of these two senses, and so obtain an asymmetrical relation of 
a to m, which, like the original relation, will have magnitude. Thus the 
case of symmetrical relations may be reduced to that of asymmetrical 
relations. These latter lead to signs, and to addition and subtraction, 
in exactly the same way as stretches with sense ; the only difference being 
that the addition, and subtraction are now of the kind which, in Part III, 
we called relational. Thus in all cases of magnitudes having sign, the 
difference between the two senses of an asymmetrical relation is the 
source of the difference of sign. 

The case which we discussed in connection with stretches is of 
fundamental importance in Geometry. We have here a magnitude with- 
out sign, an asymmetrical relation without magnitude, and some intimate 
connection between the two. The combination of both then gives a 
magnitude which has sign. All geometrical magnitudes having sign 
arise in this way. But there is a curious complication in the case of 
volumes. Volumes are, in the first instance, signless quantities ; but in 
analytical Geometry they always appear as positive or negative. Here 
the asymmetrical relations (for there are two) appear as terms, between 
which there is a symmetrical relation, but one which yet has an opposite 
of a kind very similar to the converse of an asymmetrical relation. 
This relation, as an exceptional case, must be here briefly discussed. 

222. The descriptive straight line is a serial relation in virtue of 
which the points of the line form a series*. Either sense of the descriptive 
straight line may be called a ray, the sense being indicated by an 



arrow. Any two non-coplanar rays have one or other of two relations, 
which may be called right and left -handedness respectively f. This 
relation is symmetrical but not transitive, and is the essence of the usual 
distinction of right and left. Thus the relation of the upward vertical 
to a line from north to east is right-handed, and to a line from south to 

* See Part VI. 

+ The two cases are illustrated in the figure. The diiFerence is the same as that 
between the two sorts of coordinate axes. 



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232 Order [ohap. xxyii 

east is left-handed. But though the relation is symmetrical, it is 
changed into its opposite by changing either of the terms of the relation 
into its converse. That is, denoting right-handedness by R^ left-handed- 

ness by L (which is not R)^ if A and B be two rays which are mutually 
right-handed, we shall have 

ARE, ALB, ALB, ARB, BRA, BLA, BLA, BRA, 

That is, every pair of non-coplanar straight lines gives rise to eight such 
relations, of which four are right-handed, and four left-handed. The 
diflPerence between L and R, though not, as it stands, a difference of 
sense, is, nevertheless, the difference of positive and negative, and is the 
reason why the volumes of tetrahedra, as given by determinants, always 
have signs. But there is no difficulty in following the plain man's 
reduction of right and left to asymmetrical relations. The plain man 
takes one of the rays (say ^ ) as fixed — when he is sober, he takes A to 
be the upward vertical — and then regards right and left as properties of 
the single ray B, or, what comes to the same thing, as relations of any 
two points which determine B. In this way, right and left become 
asymmetrical relations, and even have a limited degree of transitiveness, of 
the kind explained in the fifth way of generating series (in Chapter xxiv). 
It is to be observed that what is fixed must be a ray, not a mere straight 
line. For example, two planes which are not mutually perpendicular 
are not one right and the otlier left with regard to their line of inter- 
section, but only with regard to either of the rays belonging to this 
line.* But when this is borne in mind, and when we consider, not 
semi-planes, but complete planes, through the ray in question, right and 
left become asymmetrical and each other\s converses. Thus the signs 
associated with right and left, like all other signs, depend upon the 
asymmetry of relations. This conclusion, therefore, may now be allowed 
to be general. 

223. Difference of sense is, of course, more general than difference of 
sign, since it exists in cases with which mathematics (at least at present) 
is unable to deal. And difference of sign seems scarcely applicable to 
relations which are not transitive, or are not intimately connected with 
some transitive relation. It would be absurd, for example, to regard the 
relation of an event to the time of its occurrence, or of a quantity to its 
magnitude, as conferring a difference of sign. These relations are what 
Professor Schroder calls erscMpft'^', i.e. if they hold between a and h, 
they can never hold between 5 and some third term. Mathematically, 
their square is null. These relations, then, do not give rise to difference 
of sign. 

■^ This requires that the passage from the one plane to the other should he made 
via one of the acute angles made hy their intersection. 

+ Algebra der Logik, Vol. III^ p. 328. Professor Peirce calls such relations non- 
repeating (reference in Schroder^ ib.). 



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222, 223] Difference of Sense and Difference of Sign 233 

All magnitudes with sign, so the above account has led us to believe, 
are either relations or compound concepts into which relations enter. 
But what are we to say of the usual instances of opposites : good and 
evil, pleasure and pain, beauty and ugliness, desire and aversion ? The 
last pair are very complex, and if I were to attempt an analysis of them, 
I should emit some universally condemned opinions. With regard to 
the others, they seem to me to have an opposition of a very different 
kind from that of two mutually converse asymmetrical relations, and 
analogous rather to the opposition of red and blue, or of two different 
magnitudes of the same kind. From these oppositions, which are con- 
stituted by what may be called synthetic incompatibility*, the oppositions 
above mentioned differ only in the fact that there are only two incom- 
patible terms, instead of a whole series. The incompatibility consists 
in the fact that two terms which are thus incompatible cannot coexist in 
the same spatio-temporal place, or cannot be predicates of the same 
existent, or, more generally, cannot both enter into true propositions of 
a certain form, which differ only in the fact that one contains one of the 
incompatibles while the other contains the other. This kind of incom- 
patibility (which usually belongs, with respect to some class of proposi- 
tions, to the terms of a given series) is a most important notion in 
general logic, but is by no means to be identified with the difference 
between mutually converse relations. This latter is, in fact, a special 
case of such incompatibility; but it is the special case only that gives 
rise to the difference of sign. All difference of sign^ — so we may conclude 
our argument — is primarily derived from transitive asymmetrical rela- 
tions, from which it may be extended by correlation to terms variously 
related to such relations f; but such extensions are always subsequent to 
the original opposition derived from difference of sense. 

* See The Philosophy of Leibniz, by the present author (Cambridge 1900), 
pp. 19, 20. 

f Tbus in mathematical Economics_, pleasure and pain may be taken as positive 
and negative without logical error^ by the theory (whose psychological correctness 
we need not examine) that a man must be paid to endure pain^ and must pay to 
obtain pleasure. The opposition of pleasure and pain is thus correlated with that of 
money paid and money received_, which is an opposition of positive and negative in 
the sense of elementary Arithmetic. 



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CHAPTER XXVIII. 

ON THE DIFFERENCE BETWEEN OPEN AND 
CLOSED SERIES. 

224. We have now come to the end of the purely logical discussions 
concerned with order, and can turn our attention with a free mind to 
the more mathematical aspects of the subject. As the solution of the 
most ancient and respectable contradictions in the notion of infinity 
depends mainly upon a correct philosophy of order, it has been necessary 
to go into philosophical questions at some length — not so much because 
they are relevant, as because most philosophers think them so. But we 
shall reap our reward throughout the remainder of this work. 

The question to be discussed in this chapter is this: Can we ulti- 
mately distinguish open from closed series, and if so, in what does 
the distinction consist? We have seen that, mathematically, all series 
are open, in the sense that all are generated by an asymmetrical transi- 
tive relation. But philosophically, we must distinguish the different 
ways in which this relation may arise, and especially we must not 
confound the case where this relation involves no reference to other 
terms with that where such terms are essential. And practically, it is 
plain that there is some difference between open and closed series — 
between, for instance, a straight line and a circle, or a pedigree and a 
mutual admiration society. But it is not quite easy to express the 
difference precisely. 

225. Where the number of terms in the series is finite, and 
the series is generated in the first of the ways explained in 
Chapter xxiv, the method of obtaining a transitive relation out of the 
intransitive relation with which we start is radically different according 
as the series is open or closed. If R be the generating relation, and n be 
the number of terms in our series, two cases may arise. Denoting the 
relation of any term to the next but one by R^^ and so on for higher 
powers, the relation R^^ can have only one of two values, zero and 
identity. (It is assumed that i? is a one-one relation.) For starting 
with the first term, if there be one, R^'^~^ brings us to the last term ; and 
thus R'^^ gives no new term, and there is no instance of the relation 
R^^. On the other hand, it may happen that, starting with any term, 



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224, 225] Difference between Open and Closed Series 235 

R^ brings us back to that term again. These two are the only possible 
alternatives. In the first case, we call the series open ; in the second, we 
call it closed. In the first case, the series has a definite beginning and 
end ; in the second case, like the angles of a polygon, it has no peculiar 
terms. In the first case, our transitive asymmetrical relation is the 
disjunctive relation "^ power of R not greater than the (n — l)th." 
By substituting this relation, which we may call R\ for i?, our series 
becomes of the second of the six types. But in the second case no such 
simple reduction to the second type is possible. For now, the relation 
of any two terms a and m of our series may be just as well taken to be 

a power of i? as a power of J?, and the question which of any three terms 
is between the other two becomes wholly arbitrary. We might now intro- 
duce, first the relation of separation of four terms, and then the resulting 
five-term relation explained in Chapter xxv. We should then regard 
three of the terms in the five-term relation as fixed, and find that the 
resulting relation of the other two is transitive and asymmetrical. But 
here the first term of our series is wholly arbitrary, which was not the 
case before ; and the generating relation is, in reality, one of five terms,, 
not one of two. There is, however, in the case contemplated, a simpler 
method. This may be illustrated as follows : In an open series, any two 
terms a and m define two senses in which the series may be described, 
the one in which a comes before m, and the other in which m comes 
before a. We can then say of any two other terms c and g that the 
sense of the order from c to g' is the same as that of the order from 
a to m, or different, as the case may be. In this way, considering 
a and m fixed, and c and g variable, we get a transitive asymmetrical 
relation between c and g^ obtained from a transitive symmetrical relation 
of the pair c, g to the pair a, m (or m, a^ as the case may be). But this 
transitive symmetrical relation can, by the principle of abstraction, be 
analyzed into possession of a common property, which is, in this case, 
the fact that a^ m and c, g have the generating relation with the same 
sense. Thus the four-term relation is, in this case, not essential. But in a 
closed series, a and m do not define a sense of the series, even when we 
are told that a is to precede m : we can start from a and get to m in 
either direction. But if now we take a third term J, and decide that we 
are to start from a and reach m taking d on the way, then a sense of the 
series is defined. The stretch adm includes one portion of the series, but 
not the other. Thus we may go from England to New Zealand either 
by the east or by the west ; but if we are to take India on the way, we 
must go by the east. If now we consider any other term, say Z;, this 
will have some definite position in the series which starts with a and 
reaches m by way of d. In this series, y^ will come either between a and 
(7, or between d and m, or after m. Thus the three-term relation of 
a^ d^ m seems in this case sufficient to generate a perfectly definite series. 
Vailati's five-term relation will then consist in this, that with regard to 



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236 Ordei^ [chap, xxviii 

the order adm^ h coDies before (or after) any other term I of the collection. 
But it is not necessary to call in this relation in the present case, since the 
three-term relation suffices. This three-term relation may be formally 
defined as follows. There is between any two terms of our collection a 
relation which is a power of R less than the r^th. Let the relation between 
a and d be R^^ that between a and m Ry, Then if x is less than y^ we 
assign one sense to adm ; if <r is greater than «/, we assign the other. 

There will be also between a and d the relation ^^^~^^, and between a and 

m the relation R'^~y. If oc is less than «/, then n — oc'i^ greater than n — y\ 

hence the asymmetry of the two cases corresponds to that of R and R, 
The terms of the series are simply ordered by correlation with their 
numbers x and y^ those with smaller numbers preceding those with 
larger ones. Thus there is here no need of the five-term relation, every- 
thing being effected by the three-term relation, which is itself reduced to 
an asymmetrical transitive relation of two numbers. But the closed 
series is still distinguished from the open one by the fact that its first 
term is arbitrary. 

226. A very similar discussion will apply to the case where our 
series is generated by relations of three terms. To keep the analogy 
with the one-one relation of the above case, we will make the following 
assumptions. Let there be a relation B of one term to two others, and 
let the one term be called the mean, the two others the extremes. Let 
the mean be uniquely determined when the extremes are given, and let 
one extreme be uniquely determined by the mean and the other extreme. 
Further let each term that occurs as mean occur also as extreme, and 
each term that occurs as extreme (with at most two exceptions) occur 
also as mean. Finally, if there be a relation in which c is mean, and h 
and d are extremes, let there be always (except when & or (i is one of the 
two possible exceptional terms) a relation in which h is the mean and c 
one of the extremes, and another in which d is the mean and c one of the 
extremes. Then h and c will occur together in only two relations. This 
fact constitutes a relation between h and c, and only one other term 
besides h will have this new relation to c. By means of this relation, if 
there are two exceptional terms, or if, our collection being infinite, there 
is only one, we can construct an open series. If our two-term relation be 
asymmetrical, this is sufficiently evident ; but the same result can be 
proved if our two-term relation is symmetrical. For there will be at 
either end, say a, an asymmetrical relation of a to the only term which is 
the mean between a and some other term. This relation multiplied by 
the Tith power of our two-term relation, where n-\-\ is any integer less 
than the number of terms in our collection, will give a relation which 
holds between a and a number (not exceeding n-\-Y) of terms of our 
collection, of which terms one and only one is such that no number less 
than n gives a relation of a to this term. Thus we obtain a correlation 



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225-227] Difference between Open and doused Series 237 

of our terms with the natural numbers, which generates an open 
series with a for one of its ends. If, on the other hand, our collec- 
tion has no exceptional terms, but is finite, then we shall obtain 
a closed series. Let our two-term relation be P, and first suppose it 
symmetrical. (It will be symmetrical if our original three-term relation 
was symmetrical with regard to the extremes.) Then every term c of 
our collection will have the relation P to two others, which will have 
to each other the relation P'^. Of all the relations of the form P^ 
which hold between two given terms, there will be one in which m is 
least : this may be called the principal relation of our two terms. Let 
the number of terms of the collection be n. Then every term of our 
collection will have to every other a principal relation P^, where oc is 
some integer not greater than 7^/2. Given any two terms c and g of the 
collection, provided we do not have cP'^^^^g (a case which will not arise 
if n be odd), let us have cP^^g^ where oj is less than n/2. This assumption 
defines a sense of the series, which may be shown as follows. If cP^Jc^ 
where y is also less than n/^^ three cases may arise, assuming y is greater 
than a^. We may have gPy^^h^ or, if ^+^ is less than nj% we may 

n 

have gP^'^y'k^ or, ii x -{-y i^ greater than 7?/S, we may have gP"^ ^h, 
(We choose always the principal relation.) These three cases are illus- 
trated in the accompanying figure. We shall say, in these three cases, 






that, with regard to the sense cg^ (1) ^ comes after c and g^ (2) and (3) 
k comes before c and ^. If y is less than ^, and 'kP^~yg^ we shall say 
that A; is between c and ^ in the sense eg. If n is odd, this covers all 
possible cases. But if n is even, we have to consider the term c', which 
is such that cP^''^c. This term is, in a certain sense, antipodal to c ; we 
may define it as the first term in the series when the above method of 
definition is adopted. If n is odd, the first term will be that term of 
class (3) for which cP^^^~'^^'^'k. Thus the series acquires a definite order, 
but one in which, as in all closed series, the first term is arbitrary. 

227. The only remaining case is that where we start from four -term 
relations, and the generating relation has, strictly speaking, five terms. 
This is the case of projective Geometry. Here the series is necessarily 
closed ; that is, in choosing our three fixed terms for the five-term 
relation, there is never any restriction upon our choice ; and any one of 
these three may be defined to be the first. 



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238 Order [chap, xxviii 

228. Thus, to sum up : Every series being generated by a transitive 
asymmetrical relation between any two terms of the series, a series is 
open when it has either no beginning, or a beginning which is not 
arbitrary ; it is closed when it has an arbitrary beginning. Now if R 
be the constitutive relation, the beginning of the series is a term having 
the relation R but not the relation R. Whenever R is genuinely a 
two-term relation, the beginning, if it exists, must be perfectly definite. 
It is only when R involves some other term (which may be considered 
fixed) besides the two with regard to which it is transitive and asym- 
metrical (which are to be regarded as variable), that the beginning can 
be arbitrary. Hence in all cases of closed series, though there may be 
an asymmetrical one-one relation if the series is discrete, the transitive 
asymmetrical relation must be one involving one or more fixed terms 
in addition to the two variable terms with regard to which it generates 
the series. Thus although, mathematically, every closed series can be 
rendered open, and every open series closed, yet there is, in regard 
to the nature of the generating relation, a genuine distinction between 
them — a distinction, however, which is of philosophical rather than 
mathematical importance. 



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CHAPTER XXIX. 



PROGRESSIONS AND ORDINAL NUMBERS. 

229. It is now time to consider the simplest type of infinite series, 
namely that to which the natural numbers themselves belong. I shall 
postpone to the next Part all the supposed difficulties arising out of 
the infinity of such series, and concern myself here only to give the 
elementary theory of them in a form not presupposing numbers*. 

The series now to be considered are those which can be correlated, 
term for term, with the natural numbers, w^ithout requiring any change 
in the order of the terms. But since the natural numbers are a 
particular case of such series, and since the whole of Arithmetic and 
Analysis can be developed out of any one such series, without any 
appeal to number, it is better to give a definition of progressions which 
involves no appeal to number. 

A progression is a discrete series having consecutive terms, and a 
beginning but no end, and being also connected. The meaning of 
connection was explained in Chapter xxiv by means of number, but this 
explanation cannot be given now. Speaking popularly, Avhen a series 
is not connected it falls into two or more parts, each being a series 
for itself. Thus numbers and instants together form a series which 
is not connected, and so do two parallel straight lines. Whenever 
a series is originally given by means of a transitive asymmetrical rela- 
tion, we can express connection by the condition that any two terms 
of our series are to have the generating relation. But progressions 
are series of the kind that may be generated in the first of our six 
ways, namely, by an asymmetrical one-one relation. In order to pass 
from this to a transitive relation, we before employed numbers, defining 
the transitive relation as any power of the one-one relation. This 
definition will not serve now, since numbers are to be excluded. It 
is one of the triumphs of modern mathematics to have adapted an 
ancient principle to the needs of this case. 

■^ The present chapter closely follows Peano's Arithmetic. See Formulaire de 
Mathematiques, Vol. 11, § 2. I have given a mathematical treatment of the subject 
in RdM^ Vols. VII and Ylll. The subject is due^ in the main_, to Dedekind and 
Georg Cantor. 



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240 Order [chap, xxix 

The definition which we want is to be obtained from mathematical 
induction. This principle, which used to be regarded as a mere subter- 
fuge for eliciting results of which no other proof was forthcoming, has 
gradually grown in importance as the foundations of mathematics have 
been more closely investigated. It is now seen to be the principle upon 
which depend, so far as ordinals are concerned, the commutative law 
and one form of the distributive law*. This principle, which gives 
the widest possible extension to the finite, is the distinguishing mark 
of progressions. It may be stated as follows : 

Given any class of terms ^, to which belongs the first term of any 
progression, and to which belongs the term of the progression next after 
any term of the progression belonging to 5, then every term of the 
progression belongs to s. 

We may state the same principle in another form. Let ^ {x) be. 
a propositional function, which is a determinate proposition as soon 
as 00 is given. Then (j>{oc) is a function of ^', and will in general be 
true or false according to the value of x. If x be a member of a 
progression, let seq x denote the term next after x. Let (/> {x) be true 
when X is the first term of a certain progression, and let (seq x) 
be true whenever ^ {x) is true, where x is any term of the progression. 
It then follows, by the principle of mathematical induction, that (j6 {x) 
is always true if x be any term of the progression in question. 

The complete definition of a progression is as follows. Let R be 
any asymmetrical one-one relation, and u a class such that every term 
of u has the relation of R to some term also belonging to the class u. 
Let there be at least one term of the class u which does not have 
the relation R to any term of u. Let s be any class to which belongs 
at least one of the terms of u which do not have the relation R to any 
term of Uy and to which belongs also every term of u which has the 
relation R to some term belonging to both u and .9 ; and let it be such 
as to be wholly contained in any class s satisfying the above conditions. 
Then u^ considered as ordered by the relation i?, is a progression f. 

230. Of such progressions, everything relevant to finite Arithmetic 
can be proved. In the first place, we show that there can only be 
one term of u which does not have the relation R to any term of u. 
We then define the term to which x has the relation R as the successor 
of X {x being a u\ which may be written seq x. The definitions and 
properties of addition, subtraction, multiplication, division, positive and 

■^ Namely (a + fi)y = a/3 + ay. The other form^ a(/3 + 7) = a/3 + ay, holds also 
for infinite ordinal numbers^ and is thus independent of mathematical induction. 

t It should be observed that a discrete open series generated by a transitive 
relation can always be reduced,, as we saw in the preceding chapter^ to one generated 
by an asymmetrical one-one relation^ provided only that the series is finite or a 
progression. 



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229, 230] Progressions and Ordinal Numbers 241 

negative terms, and rational fractions are easily given ; and it is easily 
shown that between any two rational fractions there is always a third. 
From this point it is easy to advance to irrationals and the real 
numbers*. 

Apart from the principle of mathematical induction, what is chiefly 
interesting about this process is, that it shows that only the serial or 
ordinal properties of finite numbers are used by ordinary mathematics, 
what may be called the logical properties being^ wholly irrelevant. By 
the logical properties of numbers, I mean their definition by means of 
purely logical ideas. This process^ which has been explained in Part 11, 
may be here briefly recapitulated.'^^ We show, to begin with, that a one- 
one correlation can be effected between any two null classes, or between 
any two classes u^ v which are such that, if <^ is a u^ and oo' differs from 
zr, then ,t' cannot be a u^ with a like condition for v. The possibility 
of such one-one correlation we call similarity of the two classes w, v. 
Similarity, being symmetrical and transitive, must be analyzable (by the 
principle of abstraction) into possession of a common property. This 
we define as the number of either of the classes. When the two classes 
u^ V have the above-deflned property, we say their number is one ; and 
so on for higher numbers ; the general definition of finite numbers 
demanding mathematical induction, or the non -similarity of whole and 
part, but being always given in purely logical terms. 

It is numbers so defined that are used in daily life, and that are 
essential to any assertion of numbers. It is the fact that numbers have 
these logical properties that makes them important. But it is not 
these properties that ordinary mathematics employs, and numbers might 
be bereft of them without any injury to the truth of Arithmetic and 
Analysis. What is relevant to mathematics is solely the fact that 
finite numbers form a progression. This is the reason why mathe- 
maticians — e,g. Helmholtz, Dedekind, and Kronecker — have maintained 
that ordinal numbers are prior to cardinals ; for it is solely the ordinal 
properties of number that are relevant. But the conclusion that or- 
dinals are prior to cardinals seems to have resulted from a confusion. 
Ordinals and cardinals alike form a progression, and have exactly the 
same ordinal properties. Of either, all Arithmetic can be proved 
without any appeal to the other, the propositions being symbolically 
identical, but different in meaning. In order to prove that ordinals 
are prior to cardinals, it would be necessary to show that the cardinals 
can only be defined in terms of the ordinals. But this is false, for the 
logical definition of the cardinals is wholly independent of the ordinals f. 
There seems, in fact, to be nothing to choose, as regards logical priority, 
between ordinals and cardinals, except that the existence of the ordinals 

■^ See my article on the Logic of Relations_, RdM_, VII. 

+ Professor Peano^ who has a rare immunity from error_, has recognized this fact. 
See Formulaire, 1898, 210, note (p. 39). 

E. 16 



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242 Order [chap, xxix 

is inferred from the series of cardinals. The ordinals, as we shall see 
in the next paragraph, can be defined without any appeal to the 
cardinals ; but when defined, they are seen to imply the cardinals. 
Similarly, the cardinals can be defined without any appeal to the 
ordinals ; but they essentially form a progression, and all progressions, 
as I shall now show, necessarily imply the ordinals. 

231. The correct analysis of ordinals has been prevented hitherto by 
the prevailing prejudice against relations. People speak of a series as 
consisting of certain terms taken in a certain order, and in this idea 
there is commonly a psychological element. All sets of terms have, 
apart from psychological considerations, all orders of which they are 
capable ; that is, there are serial relations, whose fields are a given set of 
terms, which arrange those terms in any possible order. In some cases, 
one or more serial relations are specially prominent, either on account of 
their simplicity, or of their importance. Thus the order of magnitude 
among numbers, or of before and after among instants, seems emphati- 
cally the natural order, and any other seems to be artificially introduced 
by our arbitrary choice. But this is a sheer error. Omnipotence itself 
cannot give terms an order which they do not possess already: all that 
is psychological is the considei^ation of such and such an order. Thus 
when it is said that we can arrange a set of terms in any order we please, 
what is really meant is, that we can consider any of the serial relations 
whose field is the given set, and that these serial relations will give 
between them any combinations of before and after that are compatible 
with transitiveness and connection. From this it results that an order 
is not, properly speaking, a property of a given set of terms, but of a serial 
relation whose field is the given set. Given the relation, its field is given 
with it ; but given the field, the relation is by no means given. The 
notion of a set of terms in a given order is the notion of a set of terms 
considered as the field of a given serial relation ; but the consideration 
of the terms is superfluous, and that of the relation alone is quite 
sufficient. 

We may, then, regard an ordinal number as a common property of 
sets of serial relations which generate ordinally similar series. Such 
relations have what I shall call likeness^ i.e. if P, Q be two such relations, 
their fields can be so correlated term for term that two terms of which 
the first has to the second the relation P will always be correlated with 
two terms of which the first has to the second the relation Q, and 
vice versa. As in the case of cardinal numbers*, so here, we may, in 
virtue of the principle of abstraction, define the ordinal number of 
a given finite serial relation as the class of like relations. It is easy to 
show that the generating relations of progressions are all alike; the 
class of such relations will be the ordinal number of the finite integers 
in order of magnitude. When a class is finite, all series that can be 

^ Cf. § 111. 



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230-232] Pj^ogressions and Ordinal Numbers 243 

formed of its terms are ordinally similar, and are ordinally different from 
series having a different cardinal number of terms. Hence there is a 
one-one correlation of finite ordinals and cardinals, for which, as we 
shall see in Part V, there is no analogy in respect of infinite numbers. We 
may therefore define the ordinal number n as the class of serial relations 
whose domains have n terms, where tz is a finite cardinal. It is necessary, 
unless 1 is to be excluded, to take domains instead of fields here, for no 
i-elation which implies diversity can have one term in its field, though it 
may have none. This has a practical inconvenience, owing to the fact 
that 7z 4- 1 must be obtained by adding one term to the field ; but the 
point involved is one for conventions as to notation, and is quite 
destitute of philosophical importance. 

232. The above definition of ordinal numbers is direct and simple, 
but does not yield the notion of " ^zth," which would usually be regarded 
as the ordinal number. This notion is far more complex : a term is not 
intrinsically the 7ith, and does not become so by the mere specification 
of n — 1 other terms. A term is the ?^th in respect of a certain serial 
relation, when, in respect of that relation, the term in question has n — \ 
predecessors. This is the definition of "7zth," showing that this notion 
is relative, not merely to predecessors, but also to a specified serial 
relation. By induction, the various finite ordinals can be defined 
without mentioning the cardinals. A finite serial relation is one which 
is not like (in the above sense) any relation implying it but not equivalent 
to it ; and a finite ordinal is one consisting of finite serial relations. If 
?^ be a finite ordinal, n + 1 is an ordinal such that, if the last term* of 
a series of the type n -}- 1 be cut off, the remainder, in the same order, is 
of the type n. In more technical language, a serial relation of the type 
?z 4- 1 is one which, when confined to its domain instead of its field, 
becomes of the type ii. This gives by induction a definition of every 
particular finite ordinal, in which cardinals are never mentioned. Thus 
we cannot say that ordinals presuppose cardinals, though they are more 
complex, since they presuppose both serial and one-one relations, whereas 
cardinals only presuppose one-one relations. 

Of the ordinal number of the finite ordinals in order of magnitude, 
several equivalent definitions may be given. One of the simplest is, 
that this number belongs to any serial relation, which is such that any 
class contained in its field and not null has a first term, while every 
term of the series has an immediate successor, and every term except the 
first has an immediate predecessor. Here, again, cardinal numbers are 
in no way presupposed. 

Throughout the above discussions our serial relations are taken to be 
transitive, not one-one. The one-one relations are easily derived from 

^ The last term of a series (if it exists) is the term helongiiig to the converse 
domain but not to the domain of the generating- relation^ i.e. the term which is after 
but not before other terms. 

16—2 



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244 Order [chap, xxix 

the transitive ones, while the converse derivation is somewhat complicated. 
Moreover the one-one relations are only adequate to define finite series, 
and thus their use cannot be extended to the study of infinite series 
unless they are taken as derivative from the transitive ones. 

233. A few words concerning positive and negative ordinals seem to 
be here in place. If the first n terms of a progression be taken away 
{n being any finite number), the remainder still form a progression. 
With regard to the new progression, negative ordinals may be assigned 
to the terms that have been abstracted ; but for this purpose it is 
convenient to regard the beginning of the smaller progression as the 
0th term. In order to have a series giving any positive or negative 
ordinal, we need what may be called a double progression. This is a 
series such that, choosing any term x out of it, two progressions start 

from ^, the one generated by a serial relation i?, the other by R. To 
X we shall then assign the ordinal 0, and to the other terms we shall 
assign positive or negative ordinals according as they belong to the one 
or the other of the two progressions starting from x. The positive and 
negative ordinals themselves form such a double progression. They 
express essentially a relation to the arbitrarily chosen origin of the two 
progressions, and 4- n and — n express mutually converse relations. 
Thus they have all the properties which we recognize in Chapter xxvii 
as characterizing terms which have signs. 



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CHAPTER XXX. 



DEDEKIND^S THEORY OF NUMBER. 

234. The theory of progressions and of ordinal numbers, with which 
we have been occupied in the last chapter, is due in the main to two 
men — Dedekind and Cantor. Cantor's contributions, being specially 
concerned with infinity, need not be considered at present; and 
Dedekind's theory of irrationals is also to be postponed. It is his theory 
of integers of which I wish now to give an account— the theory, that is 
to say, which is contained in his " Was sind und was sollen die ZaJilenf''''^ 
In reviewing this work, I shall not adhere strictly to Dedekind's 
phraseology. He appears to have been, at the time of writing, un- 
acquainted with symbolic logic ; and although he invented as much of 
this subject as was relevant to his purpose, he naturally adopted phrases 
which were not usual, and were not always so convenient as their con- 
ventional equivalents. 

The fundamental ideas of the pamphlet in question are these *f": 
(1) the representation {Ahhildimg) of a system (^1) ; (^) the notion of a 
chain (37) ; (3) the chain of an element (44) ; (4) the generalized form 
of mathematical induction (59) ; (5) the definition of a singly infinite 
system (71). From these five notions Dedekind deduces numbers and 
ordinary Arithmetic. Let us first explain the notions, and then examine 
the deduction. 

235. (1) A representation of a class u is any law by which, to every 
term of u^ say ^, corresponds some one and only one term ^(^). No 
assumption is made, to begin with, as to whether (^{pc) belongs to the 
class u^ or as to whether <^{pc) may be the same as <^{y)^ when x and y 
are different terms of n. The definition thus amounts to this : 

A representation of a class u is a many-one relation, whose domain 
contains u^ by which terms, which may or may not also belong to u^ are 

^ 2nd ed. Brunswick, 1893 (1st ed. 1887). The principal contents of this 
hook, expressed by the Algebra of Relations, will be found in my article in RdM, 
VII, 2, 3. 

+ The numbers in brackets refer, not to pages, but to the small sections into 
which the work is divided. 



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246 Order [chap, xxx 

correlated one with each of the terms of u^. The representation is 
similar when, if oc differs from «/, both being ^^'s, then ^(x) differs from 
0(«/); that is, when the relation in question is one-one. He shows that 
similarity between classes is reflexive, symmetrical and transitive, and 
remarks (34) that classes can be classified by similarity to a given class — 
a suggestion of an idea which is fundamental in Cantor's work. 

236. {%) If there exists a relation, whether one-one or many-one, 
which correlates with a class u only terms belonging to that class, then 
this relation is said to constitute a representation of u in itself (36), 
and with respect to this relation u is called a chain (37). That is to 
say, any class u is, with respect to any many-one relation, a chain, if u is 
contained in the domain of the relation, and the correlate of a 2^ is 
always itself a it. The collection of correlates of a class is called the 
image (Bild) of the class. Thus a chain is a class whose image is 
part or the whole of itself. For the benefit of the non-mathematical 
reader, it may be not superfluous to remark that a chain with regard to 
a one-one relation, provided it has any term not belonging to the image 
of the chain, cannot be finite, for such a chain must contain the same 
number of terms as a proper part of itself f. 

237. (3) If a be any term or collection of terms, there may be, 
with respect to a given many-one relation, many chains in which a is 
contained. The common part of all these chains, which is denoted by ^o? 
is what Dedekind calls the chain of a (44). For example, if a be the 
number ?2, or any set of numbers of which n is the least, the chain of a 
with regard to the relation "less by l'' will be all numbers not less 
than n. 

238. (4) Dedekind now proceeds (59) to a theorem which is 
a generalized form of mathematical induction. This theorem is as 
follows : Let a' be any term or set of terms contained in a class ,?, and let 
the image of the common part of s and the chain of a be also contained 
in s ; then it follows that the chain of a is contained in s. This some- 
what complicated theorem may become clearer by being put in other 
language. Let us call the relation by which the chain is generated (or 
rather the converse of this relation) succession, so that the correlate or 
image of a term will be its successor. Let ^ be a term which has a 
successor, or a collection of such terms. A chain in general (with regard 
to succession) will be any set of terms such that the successor of any 
one of them also belongs to the set. The chain of a will be the common 



* A many-one relation is one in wliicli_, as in tlie relation of a quantity to its 
magnitude^ the right-hand term^ to which the relation is^ is uniquely determined 
when the left-hand term is given. Whether the converse holds is left undecided. 
Thus a one-one relation is a particular case of a many-one relation. 

t A proper part (Echter Theil) is a phrase analogous to ^^ proper fraction" ; it 
means a part not the whole. 



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235-240] Dedekind's Theory of Number 247 

part of all the chains containing a. Then the data of the theorem 
inform us that a is contained in s^ and, if any term of the chain of a be 
an s^ so is its successor ; and the conclusion is, that every term in the 
chain of a is an s. This theorem, as is evident, is very similar to 
mathematical induction, from which it differs^ first by the fact that a 
need not be a single term, secondly by the fact that the constitutive 
relation need not be one-one, but may be many-one. It is a most 
remarkable fact that Dedekind's previous assumptions suffice to demon- 
strate this theorem. 

239. (5) I come next to the definition of a singly infinite system 
or class (71). This is defined as a class which can be represented in 
itself by means of a one-one relation, and which is further such as to be 
the chain, with regard to this one-one relation, of a single term of the 
class not contained in the image of the class. Calling the class iV, and 
the one-one relation R^ there are, as Dedekind remarks, four points in 
this definition. (1) The image of N is contained in iV; that is, every 
term to which an N has the relation R is an N. (S) N is the chain of 
one of its terms. (3) This one term is such that no N has the relation 
R to it, i.e. it is not the image of any other term of N. (4) The 
relation R is one-one, in other words, the representation is similar. The 
abstract system, defined simply as possessing these properties, is defined 
by Dedekind as the ordinal numbers (73). It is evident that his singly 
infinite system is the same as what we called a progression, and he 
proceeds to deduce the various properties of progressions, in particular 
mathematical induction (80), which follows from the above generalized 
form. One number m is said to be less than another n, when the chain 
of n is contained in the image of the chain of m (89) ; and it is shown 
(88, 90) that of two diff'erent numbers, one must be the less. From this 
point everything proceeds simply. 

240. The only further point that seems important for our present 
purpose is the definition of cardinals. It is shown (13^) that all singly 
infinite systems are similar to each other and to the ordinals, and that 
conversely (133) any system which is similar to a singly infinite system 
is singly infinite. When a system is finite, it is similar to some system 
Z^, where Z^ means all the numbers from 1 to n both inclusive ; and 
vice versa (160). There is only one number 7i which has this property 
in regard to any given finite system, and when considered in relation 
to this property it is called a cardinal number, and is said to be the 
number of elements of which the said system consists (161). Here 
at last we reach the cardinal numbers. Their dependence on ordinals, 
if I may venture to interpret Dedekind, is as follows : owing to the 
order of the ordinals, every ordinal n defines a class of ordinals Z^, 
consisting of all that do not succeed it. They may be defined as all 
that are not contained in the image of the chain of n. This class of 
ordinals may be similar to another class^ which is then said to have the 



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248 Order [chap, xxx 

cardinal number n. But it is only because of the order of the ordinals 
that each of them defines a class, and thus this order is presupposed in 
obtaining cardinals. 

241. Of the merits of the above deduction it is not necessary for 
me to speak, for they are universally acknowledged. But some points 
call for discussion. In the first case, Dedekind proves mathematical 
induction, while Peano regards it as an axiom. This gives Dedekind 
an apparent superiority, which must be examined. In the second place, 
there is no reason, merely because the numbers which Dedekind obtains 
have an order, to hold that they are ordinal numbers ; in the third 

' place, his definition of cardinals is unnecessarily complicated, and the 
dependence of cardinals upon order is only apparent. I shall take these 
points in turn. 

As regards the proof of mathematical induction, it is to be observed 
that it makes the practically equivalent assumption that numbers form 
the chain of one of them. Either can be deduced from the other, and 
the choice as to which is to be an axiom, which a theorem, is mainly 
a matter of taste. On the whole, though the consideration of chains 
is most ingenious, it is somewhat difficult, and has the disadvantage 
that theorems concerning the finite class of numbers not greater than n 
as a rule have to be deduced from corresponding theorems concerning 
the infinite class of numbers greater than n. For these reasons, and 
not because of any logical superiority, it seems simpler to begin with 
mathematical induction. And it should be observed that, in Peano's 
method, it is only when theorems are to be proved concerning any 
number that mathematical induction is required. The elementary 
Arithmetic of our childhood, which discusses only particular numbers, 
is wholly independent of mathematical induction ; though to prove that 
this is so for every particular number would itself require mathematical 
induction. In Dedekind's method, on the other hand, propositions 
concerning particular numbers, like general propositions, demand the 
consideration of chains. Thus there is, in Peano's method, a distinct 
advantage of simplicity, and a clearer separation between the particular 
and the general propositions of Arithmetic. But from a purely logical 
point of view, the two methods seem equally sound; and it is to be 
remembered that, with the logical theory of cardi^nals, both Peano's and 
Dedekind's axioms become demonstrable*. 

242. On the second point, there is some deficiency of clearness in 
what Dedekind says. His words are (73): "If in the contemplation 
of a singly infinite system iV, ordered by a representation (/>, we disregard 
entirely the peculiar nature of the elements, retaining only the possibility 
of distinguishing them, and considering only the relations in which they 
are placed by the ordering representation ^, then these elements are 
called natural numbers or ordinal numbers or simply numbers^ Now 

^ Cf. Chap. XIII. 



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240-243] Dedekind's Theory of Number 249 

it is impossible that this account should be quite correct. For it implies 
that the terms of all progressions other than the ordinals are complex, 
and that the ordinals are elements in all such terms, obtainable by 
abstraction. But this is plainly not the case. A progression can be 
formed of points or instants, or of transfinite ordinals, or of cardinals, 
in which, as we shall shortly see, the ordinals are not elements. More- 
over it is impossible that the ordinals should be, as Dedekind suggests, 
nothing but the terms of such relations as constitute a progression. 
If they are to be anything at all, they must be intrinsically something ; 
they must differ from other entities as points from instants, or colours 
from sounds. yWhat Dedekind intended to indicate was probably a 
definition by means of the principle of abstraction, such as we attempted 
to give in the preceding chapter. But a definition so made always 
indicates some class of entities having (or being) a genuine nature of 
their own, and not logically dependent upon the manner in which they 
have been defined. The entities defined should be visible, at least to 
the mind's eye ; what the principle asserts is that, under certain con- 
ditions, there are such entities, if only we knew where to look for them. 
But whether, when we have found them, they will be ordinals or 
cardinals, or even something quite different, is not to be decided 
off-hand. And in any case, Dedekind does not show us what it is 
that all progressions have in common, nor give any reason for supposing 
it to be the ordinal numbers, except that all progressions obey the same 
laws as ordinals do, which would prove equally that any assigned 
progression is what all progressions have in common. 

243. This brings us to the third point, namely the definition of 
cardinals by means of ordinals. Dedekind remarks in his preface (p. ix) 
that many will not recognize their old friends the natural numbers in 
the shadowy shapes which he introduces to them. In this, it seems 
to me, the supposed persons are in the right — in other words, I am one 
among them. What Dedekind presents to us is not the numbers, 
but any progression : what he says is true of all progressions alike, 
and his demonstrations nowhere — not even where he comes to cardinals — 
involve any property distinguishing numbers from other progressions. 
No evidence is brought forward to show that numbers are prior to 
other progressions. We are told, indeed, that they are what all pro- 
gressions have in common ; but no reason is given for thinking that 
progressions have anything in common beyond the properties assigned 
in the definition, which do not themselves constitute a new progression. 
The fact is that all depends upon one-one relations, which Dedekind 
has been using throughout without perceiving that they alone suffice 
for the definition of cardinals. The relation of similarity between 
classes, which he employs consciously, combined with the principle of 
abstraction^ which he implicitly assumes, suffice for the definition of 
cardinals ; for the definition of ordinals these do not suffice ; we 



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250 Order [chap, xxx 

require, as we saw in the preceding chapter, the relation of Hkeness 
between well-ordered serial relations. The definition of particular 
finite ordinals is effected explicitly in terms of the corresponding 
cardinals : if 7^ be a finite cardinal number, the ordinal number n is 
the class of serial relations which have n terms in their domain 
(or in their field, if we prefer this definition). In order to define 
the notion of " nth,"' we need, besides the ordinal number ?z, the 
notion of powers of a relation, i.e. of the relative product of a rela- 
tion multiplied into itself a finite number of times. Thus if R be any 
one-one serial relation, generating a finite series or a progression, the first 
term of the field of R (which field we will call r) is the term belonging 
to the domain, but not to the converse domain, i.e.^ having the relation 
R but not the relation R. If r has n or more terms, where n is a finite 
number, the ?ith term of r is the term to which the first term has the 
relation i?^~^, or, again, it is the term having the relation R'*^"^ but not 
the relation R^^. Through the notion of powers of a relation, the 
introduction of cardinals is here unavoidable ; and as powers are defined 
by mathematical induction, the notion of 72th, according to the above 
definition, cannot be extended beyond finite numbers. We can however 
extend the notion by the following definition : If P be a transitive 
aliorelative generating a well-ordered series p, the n\h term of p is the 
term x such that, if P' be the relation P limited to x and its pre- 
decessors, then P' has the ordinal number n. Here the dependence 
upon cardinals results from the fact that the ordinal n can, in general, 
only be defined by means of the cardinal n. 

It is important to observe that no set of terms has inherently one 
order rather than another, and that no term is the n\h of a set except 
in relation to a particular generating relation whose field is the set or 
part of the set. For example, since in any progression, any finite 
number of consecutive terms including the first may be taken away, 
and the remainder will still form a progression, the ordinal number 
of a term in a progression may be diminished to any smaller number 
we choose. Thus the ordinal number of a term is relative to the series 
to which it belongs. This may be reduced to a relation to the first 
term of the series ; and lest a vicious circle should be suspected, it may 
be explained that the j^r^^ term can always be defined non-numerically. 
It is, in Dedekind's singly infinite system, the only term not contained 
in the image of the system ; and generally, in any series, it is the only 
term which has the constitutive relation with one sense, but not with 
the other*. Thus the relation expressed by nth. is not only a relation 
to 7i, but also to the first term of the series ; and Jirst itself depends 

■^ Though when the series has two ends^ we have to make an arbitrary selection 
as to which we will call lirst^ which last. The obviously non-numerical nature of 
last illustrates that of its correlative^ first. 



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243] DedeMnd's Theory of Number 251 

upon the terms included in the series, and upon the relation by which 
they are ordered, so that what was first may cease to be so, and what 
was not first may become so. Thus the first term of a series must be 
assigned, as is done in Dedekind'^s view of a progression as the chain 
of its first term. Hence n\h expresses a four-cornered relation, between 
the term which is ?ith, an assigned term (the first), a generating serial 
relation, and the cardinal number n. Thus it is plain that ordinals, 
either as classes of like serial relations, or as notions like " n\h^'' are 
more complex than cardinals ; that the logical theory of cardinals is 
wholly independent of the general theory of progressions, requiring 
independent development in order to show that the cardinals form a 
progression ; and that Dedekind^s ordinals are not essentially either 
ordinals or cardinals, but the members of any progression whatever. 
I have dwelt on this point, as it is important, and my opinion is at 
variance with that of most of the best authorities. If Dedekind's view 
were correct, it would have been a logical error to begin, as this work 
does, with the theory of cardinal numbers rather than with order. 
For my part, I do not hold it an absolute error to begin with order, 
since the properties of progressions, and even most of the properties of 
series in general, seem to be largely independent of number. But 
the properties of number must be capable of proof without appeal to 
the general properties of progressions, since cardinal numbers can be 
independently defined, and must be seen to form a progression before 
theorems concerning progressions can be applied to them. Hence the 
question, whether to begin with order or with numbers, resolves itself 
into one of convenience and simplicity ; and from this point of view, 
the cardinal numbers seem naturally to precede the very difficult con- 
siderations as to series which have occupied us in the present Part. 



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CHAPTER XXXI. 



DISTANCE. 

244. The notion of distance is one which is often supposed essential 
to series*, but which seldom receives precise definition. An emphasis on 
distance characterizes, generally speaking, those who believe in relative 
position. Thus Leibniz, in the course of his controversy with Clarke, 
remarks : 

"As for the objection, that space and time are quantities, or rather 
things endowed with quantity, and that situation and order are not 
so: I answer, that order also has its quantity; there is that in it which 
goes before, and that which follows; there is distance or interval. 
Relative things have their quantity, as well as absolute ones. For 
instance, ratios or proportions in mathematics have their quantity, and 
are measured by logarithms; and yet they are relations. And therefore, 
though time and space consist in relations, yet they have their 
quantity f.'' 

In this passage, the remark : " There is that which goes before, and 
that which follows ; there is distance or interval,^' if considered as an 
inference, is a non sequitur; the mere fact of order does not prove that 
there is distance or interval. It proves, as we have seen, that there 
are stretches, that these are capable of a special form of addition 
closely analogous to what I have called relational addition, that they 
have sign, and that (theoretically at least) stretches which fulfil the 
axioms of Archimedes and of linearity are always capable of numerical 
measurement. But the idea, as Meinong rightly points out, is entirely 
distinct from that of stretch. Whether any particular series does or 
does not contain distances, will be, in most compact series {ix. such as 
have a term between any two), a question not to be decided by argument. 
In discrete series there must be distance ; in others, there may be — 
unless, indeed, they are series obtained from progressions as the 
rationals or the real numbers are obtained from the integers, in which 

■^ E.g. by Meinong^ op. cit. § 17. 

t Phil. Werke, Gerhardt's ed. Vol. vii^ p. 404. 



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244, 245] Distance 253 

case there must be distance. But we shall find that stretches are mathe- 
matically sufficient, and that distances are complicated and unimportant. 

245. The definition of distance, to begin with, is no easy matter. 
What has been done hitherto towards this end is chiefly due to non- 
Euclidean Geometry*; something also has been done towards settling the 
definition by Meinong-f*. But in both these cases, there is more concern for 
numerical measurement of distance than for its actual definition. Never- 
theless, distance is by no means indefinable. Let us endeavour to genera- 
lize the notion as much as possible. In the first place, distance need not 
be asymmetrical; but the other properties of distance always allow us to 
render it so, and we may therefore take it to be so. Secondly, a distance 
need not be a quantity or a magnitude ; although it is usually taken to 
be such, we shall find the taking it so to be irrelevant to its other 
properties, and in particular to its numerical measurement. Thirdly, 
when distance is taken asymmetrically, there must be only one term to 
which a given term has a given distance, and the converse relation to the 
given distance must be a distance of the same kind. (It will be observed 
that we must first define a hind of distance, and proceed thence to the 
general definition of distance.) Thus every distance is a one-one 
relation; and in respect to such relations it is convenient to respect the 
converse of a relation as its -1th power. Further the relative product 
of two distances of a kind must be a distance of the same kind. When 
the two distances are mutually converse, their product will be identity, 
which is thus one among distances (their zero, in fact), and must be the 
only one which is not asymmetrical. Again the product of two distances 
of a kind must be commutative |. If the distances of a kind be magni- 
tudes, they must form a kind of magnitude — i.e, any two must be equal 
or unequal. If they are not magnitudes, they must still form a series 
generated in the second of our six ways, i.e. every pair of different 
distances must have a certain asymmetrical relation, the same for all 
pairs except as regards sense. And finally, if Q be this relation, and 
Ri QR2 (^1? ^2? being distances of the kind), then if R^ be any other 
distance of the kind, we must have R^ R^ QR^ R^. All these properties, 
so far as I can discover, are independent ; and we ought to add a 
property of the field, namely this : any two terms, each of which belongs 
to the field of some distance of the kind (not necessarily the same for 
both), have a relation which is a distance of the kind. Having now 
defined a kind of distance, a distance is any relation belonging to some 
kind of distance; and thus the work of definition seems completed. 

The notion of distance, it will be seen, is enormously complex. The 
properties of distances are analogous to those of stretches with sign, but 

■^ See e.g. Whiteliead_, Unimi'sal Algebra, Cambridge^ 1898^ Book vi_, Chap. i. 
+ Op. cit. Section iv. 

X This is an independent property ; consider for instance the difference between 
^^^maternal grandfather" and '^'^ paternal grandmother." 



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254 Order [chap, xxxi 

are far less capable of mutual deduction. The properties of stretches 
corresponding to many of the above properties of distances are capable 
of proof. The difference is largely due to the fact that stretches can be 
added in the elementary logical (not arithmetical) way, whereas distances 
require what I have called relational addition, which is much the same as 
relative multiplication. 

246. The numerical measurement of distances has already been par- 
tially explained in Part III. It requires, as we saw, for its full application, 
two further postulates, which, however, do not belong to the definition of 
distances, but to certain kinds of distances only. These are, the postu- 
late of Archimedes : given any two distances of a kind, there exists 
a finite integer n such that the n\h power of the first distance is greater 
than the second distance; and Du Bois Reymond's postulate of linearity: 
Any distance has an n\h root, where n is any integer (or any prime, 
whence the result follows for any integer). When these two postulates 
are satisfied, we can find a meaning for i?^, where i? is a distance of the 
kind other than identity, and oo is any real number*. Moreover, any 
distance of the kind is of the form i?*', for some value of oc. And oc is, 
of course, the numerical measure of the distance. 

In the case of series generated in the first of our six ways, the various 
powers of the generating relation R give the distances of terms. These 
various powers, as the reader can see for himself, verify all the above 
characteristics of ' distances. In the case of series generated from pro- 
gressions as rationals or real numbers from integers, there are always 
distances; thus in the case of the rationals themselves, which are one- 
one relations, their differences, which are again rationals, measure or 
indicate relations between them, and these relations are of the nature of 
distances. And we shall see, in Part V, that these distances have some 
importance in connection with limits. For numerical measurement in 
some form is essential to certain theorems about limits, and the nume- 
rical measurement of distances is apt to be more practically feasible than 
that of stretches. 

247. On the general question, however, whether series unconnected 
with number — for instance spatial and temporal series — are such as to 

. contain distances, it is difficult to speak positively. Some things may 
be said against this view. In the first place, there must be stretches, and 
these must be magnitudes. It then becomes a sheer assumption — which 
must be set up as an axiom — that equal stretches correspond to equal 
distances. This may, of course, be denied, and we might even seek an 

■^ The powers of distances are here understood in the sense resulting from relative 
multiplication ; thus if a and h have the same distance as h and c, this distance is the 
square root of the distance of a and c. The postulate of linearity_, whose expression 
in ordinary language is: '^'^ every linear quantity can be divided into n equal parts,, 
where n is any integer/' will be found in Du Bois Reymond's AUgemeine Functionen- 
theorie (Tubingen, 1882), p. 46. 



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245-248] Distance 255 

interpretation of non-Euclidean Geometry in the denial. We might 
regard the usual coordinates as expressing stretches, and the logarithms 
of their anharmonic ratios as expressing distances ; hyperbolic Geometry, 
at least, might thus find a somewhat curious interpretation. Herr 
Meinong, who regards all series as containing distances, maintains an 
analogous principle with regard to distance and stretch in general. The 
distance, he thinks, increases only as the logarithm of the stretch. It 
may be observed that, where the distance itself is a rational number 
(w^hich is possible, since rationals are one-one relations), the opposite 
theory can be made formally convenient by the following fact. The 
square of a distance, as we saw generally, is said to be twice as great as 
the distance whose square it is. We might, where the distance is a 
rational, say instead that the stretcli is twice as great, but that the 
distance is truly the square of the former distance. For where the 
distance is already numerical, the usual interpretation of numerical 
measurement conflicts with the notation R". Thus we shall be com- 
pelled to regard the stretch as proportional to the logarithm of the 
distance. But since, outside the theory of progressions, it is usually 
doubtful whether there are distances, and since, in almost all other 
series, stretches seem adequate for all the results that are obtainable, the 
retention of distance adds a complication for which, as a rule, no 
necessity appears. It is therefore generally better, at least in a philo- 
sophy of mathematics, to eschew distances except in the theory of 
progressions, and to measure them, in that theory, merely by the 
indices of the powers of the generating relation. There is no logical 
reason, so far as I know, to suppose that there are distances elsewhere, 
except in a finite space of two dimensions and in a projective space ; and 
if there are, they are not mathematically important. We shall see in 
Part VI how the theory of space and time may be developed without pre- 
supposing distance; the distances which appear in projective Geometry are 
derivative relations, not required in defining the properties of our space ; 
and in Part V we shall see how few are the functions of distance with 
regard to series in general. And as against distance it may be remarked 
that, if every series must contain distances, an endless regress becomes 
unavoidable, since every kind of distance is itself a series. This is not, 
I think, a logical objection, since the regress is of the logically permis- 
sible kind ; but it shows that great complications are introduced by 
regarding distances as essential to every series. On the whole, then, it 
seems doubtful whether distances in general exist; and if they do, their 
existence seems unimportant and a source of very great complications. 

248. We have now completed our review of order, in so far as is 
possible without introducing the difficulties of continuity and infinity. 
We have seen that all order involves asymmetrical transitive relations, and 
that every series as such is open. But closed series, we found, could be 
distinguished by the mode of their generation, and by the fact that, 



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256 Oi^der [chap, xxxi 

though they always have a first term, this term may always be selected 
arbitrarily. We saw that asymmetrical relations must be sometimes 
unanalyzable, and that when analyzable, other asymmetrical relations 
must appear in the analysis. The difference of sign, we found, depends 
always upon the difference between an asymmetrical relation and its 
converse. In discussing the particular type of series which we called 
progressions, we saw how all Arithmetic applies to every such series, and 
how finite ordinals may be defined by means of them. But though we 
found this theory to be to a certain extent independent of the cardinals, 
we saw no reason to agree with Dfedekind in regarding cardinals as 
logically subsequent to ordinals. Finally, we agreed that distance is 
a notion which is not essential to series, and of little importance outside 
Arithmetic. With this equipment, we shall be able, I hope, to dispose 
of all the difficulties which philosophers have usually found in infinity 
and continuity. If this can be accomplished, one of the greatest of 
philosophical problems will have been solved. To this problem Part V 
is to be devoted. 



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PART V. 

INFINITY AND CONTINUITY. 



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CHAPTER XXXII. 



THE CORRELATION OF SERIES. 

249. We come now to what has been generally considered the 
fundamental problem of mathematical philosophy — I mean, the problem 
of infinity and continuity. This problem has undergone, through the 
labours of Weierstrass and Cantor, a complete transformation. Since 
t\ie time of Newton and Leibniz, the nature of infinity and continuity 
had been sought in discussions of the so-called Infinitesimal Calculus. 
But it has been shown that this Calculus is not, as a matter of fact, 
in any way concerned with the infinitesimal, and that a large and most 
important branch of mathematics is logically prior to it. The problem 
of continuity, moreover, has been to a great extent separated from that 
of infinity. It was formerly supposed — and herein lay the real strength 
of Kant's mathematical philosophy — that continuity had an essential 
reference to space and time, and that the Calculus (as the word fluxion 
suggests) in some way presupposed motion or at least change. In this 
view, the philosophy of space and time was prior to that of continuity, 
the Transcendental Aesthetic preceded the Transcendental Dialectic, and 
the antinomies (at least the mathematical ones) were essentially spatio- 
temporal. All this has been changed by modern mathematics. What 
is called the arithmetization of mathematics has shown that all the 
problems presented, in this respect, by space and time, are already 
present in pure arithmetic. The theory of infinity has two forms, 
cardinal and ordinal, of which the former springs from the logical 
theory of number ; the theory of continuity is purely ordinal. In the 
theory of continuity and the ordinal theory of infinity, the problems 
that arise are not specially concerned with numbers, but with all series 
of certain types which occur in arithmetic and geometry alike. What 
makes the problems in question peculiarly easy to deal with in the case 
of numbers is, that the series of rationals, which is what I shall call a 
compact series, arises from a progression, namely that of the integers, and 
that this fact enables us to give a proper name to every term of the 
series of rationals — a point in which this series differs from others of the 
same type. But theorems of the kind which will occupy us in most of 

17—2 



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260 Infinity and Continuity [chap, xxxii 

the following chapters, though obtained in arithmetic, have a fair wider 
application, since they are purely ordinal, and involve none of the 
logical properties of numbers. That is to say, the idea whic:h the 
Germans call Anzahl^ the idea of the number of terms in some class, 
is irrelevant, save only in the theory of transfinite cardinals- -an 
important but very distinct part of Cantor's contributions to the tht^ory 
of infinity. We shall find it possible to give a general definition of 
continuity, in which no appeal is made to the mass of unanalj^zed 
prejudice which Kantians call "intuition"; and in Part VI we shall 
find that no other continuity is involved in space and time. And we 
shall find that, by a strict adherence to the doctrine of limits, ii: is 
possible to dispense entirely with the infinitesimal, even in the definition 
of continuity and the foundations of the Calculus. 

250. It is a singular fact that, in proportion as the infinitesimal 
has been extruded from mathematics, the infinite has been allowed 
a freer development. From Cantor's work it appears that there are 
two respects in which infinite numbers differ from those that are finite. 
The first, which applies to both cardinals and ordinals, is, that they do 
not obey mathematical induction — or rather, they do not form part of 
a series of numbers beginning with 1 or 0, proceeding in order of 
magnitude, containing all numbers intermediate in magnitude between 
any two of its terms, and obeying mathematical induction. The 
second, which applies only to cardinals, is, that a whole of an infinite 
number of terms always contains a part consisting of the same 
number of terms. The first respect constitutes the true definition 
of an infinite series, or rather of what we may call an infinite 
term in a series : it gives the essence of the ordinal infinite. The 
second gives the definition of an infinite collection, and will doubtless 
be pronounced by the philosopher to be plainly self-contradictory. But 
if he will condescend to attempt to exhibit the contradiction, he will 
find that it can only be proved by admitting mathematical induction, 
so that he has merely established a connection with the ordinal infinite. 
Thus he will be compelled to maintain that the denial of mathematical 
induction is self -contradictory ; and as he has probably reflected little, 
if at all, on this subject, he will do well to examine the matter before 
pronouncing judgment. And when it is admitted that mathematical 
induction may be denied without contradiction, the supposed antinomies 
of infinity and continuity one and all disappear. This I shall endeavour 
to prove in detail in the following chapters. 

251. Throughout this Part we shall often have occasion for a 
notion which has hitherto been scarcely mentioned, namely the correla- 
tion of series. In the preceding Part we examined the nature of 
isolated series, but we scarcely considered the relations between different 
series. These relations, however, are of an importance which philo- 
sophers have wholly overlooked, and mathematicians have but lately 



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249-251] The Correlation of Series 261 

realized. It has long been known how much could be done in Geometry 
by means of homography, which is an example of correlation ; and it 
has been shown by Cantor how important it is to know whether a series 
is denumerable, and how similar two series capable of correlation are. 
But it is not usually pointed out that a dependent variable and its 
independent variable are, in most mathematical cases, merely correlated 
series, nor has the general idea of correlation been adequately dealt 
with. In the present work only the philosophical aspects of the subject 
are relevant. 

Two seiies s^ s' are said to be correlated when there is a one-one 
relation R coupling every term of s with a term of s\ and vice vei^sd^ and 
when, if oc^ y be terms of s^ and x precedes «/, then their correlates x\ y in 
/ are such that x precedes y' , Two classes or collections are correlated 
whenever there is a one-one relation between the terms of the one and 
the terms of the other, none being left over. Thus two series may be 
correlated as classes without being correlated as series ; for correlation 
as classes involves only the same cardinal number, whereas correlation 
as series involves also the same ordinal type — a distinction whose 
importance will be explained hereafter. In order to distinguish these 
cases, it will be well to speak of the correlation of classes as correlation 
simply, and of the correlation of series as ordinal correlation. Thus 
whenever correlation is mentioned without an adjective, it is to be 
understood as being not necessarily ordinal. Correlated classes will be 
called similar ; correlated series will be called ordinally similar ; and 
their generating relations will be said to have the relation of 
liheness. 

Correlation is a method by which, when one series is given, others 
may be generated. If there be any series whose generating relation 
is P, and any one-one relation which holds between any term x of the 
series and some term which we may call xji^ then the class of terms 
xji will form a series of the same type as the class of terms x. For 
suppose y to be any other term of our original series, and assume xPy. 
Then we have Xj^Rx^ ^^y^ and yRyn. Hence xj^RFRy^. Now it may 
be shown* that, if P be transitive and asymmetrical, so is RFR\ hence 
the correlates of terms of the P-series form a series whose generating 
relation is RPR. Between these two series there is ordinal correlation, 
and the series have complete ordinal similarity. In this way a new 
series, similar to the original one, is generated by any one-one relation 
whose field includes the original series. It can also be shown that, 
conversely, if P, P' be the generating relations of two similar series, 
there is a one-one relation J?, whose domain is the field of P, which 
is such that P'=5Pi?. 

^ See my article in RdM_, Vol. viii_, No. 2. 



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262 Infinity and Contiimity [chap, xxxii 

252. We can now understand a distinction of great importance, 
namely that between self-sufficient or independent series, and series by 
correlation. In the case just explained there is perfect mathematical 
symmetry between the original series and the series by correlation ; for, if 
we denote by Q the relation RPR^ we shall find P — RQR. Thus we may 
take either the Q-series or the P-series as the original, and regard the 
other as derivative. But if it should happen that J?, instead of being 
one-one, is many-one, the terms of the field of Q, which we will call g, 
will have an order in which there is repetition, the same term occurring 
in different positions corresponding to its different correlates in the field 
of P, which we will call p. This is the ordinary case of mathematical 
functions which are not linear. It is owing to preoccupation with such 
series that most mathematicians fail to realize the impossibility, in an 
independent series, of any recurrence of the same term. In every 
sentence of print, for example, the letters acquire an order by correlation 
with the points of space, and the same letter will be repeated in different 
positions. Here the series of letters is essentially derivative, for we 
cannot order the points of space by relation to the letters : this would 
give us several points in the same position, instead of one letter in several 
positions. In fact, if P be a serial relation, and i? be a many-one relation 
whose domain is the field of P, and Q == RPR, then Q has all the character- 
istics of a serial relation except that of implying diversity ; but RQR is 
not equivalent to P, and thus there is a lack of symmetry. It is for 
this reason that inverse functions in mathematics, such as sin~^ ^, are 
genuinely distinct from direct functions, and require some device or 
convention before they become unambiguous. Series obtained from 
a many-one correlation as q was obtained above will be called series 
by correlation. They are not genuine series, and it is highly important 
to eliminate them from discussions of fundamental points. 

253. The notion of likeness corresponds, among relations, to similarity 
among classes. It is defined as follows : Two relations P, Q are like 
when there is a one-one relation S such that the domain of S is the field 
of P, and Q = SPS. This notion is not confined to serial relations, but 
may be extended to all relations. We may define the relation-nimiber 
of a relation P as the class of all relations that are like P; and we can 
proceed to a very general subject which may be called relation-arithmetic. 
Concerning relation -numbers we can prove those of the formal laws of 
addition and multiplication that hold for transfinite ordinals, and thus 
obtain an extension of a part of ordinal arithmetic to relations in 
general. By means of likeness we can define a finite relation as one 
which is not like any proper part of itself — a proper part of a relation 
being a relation which implies it but is not equivalent to it. In this 
way we can completely emancipate ourselves from cardinal arithmetic. 
Moreover the properties of likeness are in themselves interesting and 



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252-254] The Correlation of Series 263 

important. One curious property is that, if S be one-one and have the 
field of P for its domain, the above equation Q = SPS is equivalent to 
SQ = PS or to QS= SP*. 

254. Since the correlation of series constitutes most of the mathe- 
matical examples of functions, and since function is a notion which is 
not often clearly explained, it will be well at this point to say something 
concerning the nature of this notion. In its most general form, function- 
ality does not differ from relation. For the present purpose it will be 
well to recall two technical terms, which were defined in Part I. If a^ 
has a certain relation to ?/, I shall call ^ the referent^ and y the relatum^ 
with regard to the relation in question. If now oc be defined as belonging 
to some class contained in the domain of the relation, then the relation 
defines «/ as a function of x. That is to say, an independent variable 
is constituted by a collection of terms, each of which can be referent 
in regard to a certain relation. Then each of these terms has one or 
more relata, and any one of these is a certain function of its referent, 
the function being defined by the relation. Thus father defines a 
function, provided the independent variable be a class contained in that 
of male animals who have or will have propagated their kind ; and 
if A be the father of 5, B is said to be a function of A, What is 
essential is an independent variable, i.e. any term of some class, and 
a relation whose extension includes the variable. Then the referent 
is the independent variable, and its function is any one of the cor- 
responding relata. 

But this most general idea of a function is of little use in mathematics. 
There are two principal ways of particularizing the function : first, we 
may confine the relations to be considered to such as are one-one or 
many-one, i.e. such as give to every referent a unique relattiiEft ; secondly, 
we may confine the independent variable to series. The second par- 
ticularization is very important, and is specially relevant to our present 
topics. But as it almost wholly excludes functions from Symbolic 
Logic, where series have little importance, we may as well postpone it for 
a moment while we consider the first particularization alone. 

The idea of function is so important, and has been so often con- 
sidered with exclusive reference to numbers, that it is well to fill our 
minds with instances of n on -numerical functions. Thus a very important 
class of functions are propositions containing a variable f. Let there be 
some proposition in which the phrase " any «" occurs, where a is some 
class. Then in place of "any a'" we may put ^, where x is an undefined 
member of the class a — in other words, any a. The proposition then 
becomes a function of x^ which is unique when x is given. This pro- 
position will, in general, be true for some values of x and false for others. 

* On this subject see my article in RdM, Vol. viii, especially Nos. 2, 6. 
t These are what in Part I we called prepositional functions. 



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264 Infinity and Continuity [chap, xxxii 

The values for which the function is true form what might be called, 
by analogy with Analytic Geometry, a logical curve. This general 
view may, in fact, be made to include that of Analytic Geometry. The 
equation of a plane curve, for example, is a propositional function which 
is a function of two variables oc and «/, and the curve is the assemblage of 
points which give to the variables values that make the proposition true. 
A proposition containing the word any is the assertion that a certain 
propositional function is true for all values of the variable for which it is 
significant. Thus " any man is mortal '''' asserts that " x is a man implies 
^ is a mortal"' is true for all values of x for which it is significant, which 
may be called the admissible values. Propositional functions, such as 
"^ is a number,"' have the peculiarity that they look like propositions, 
and seem capable of implying other propositional functions, while yet 
they are neither true nor false. The fact is, they are propositions for all 
admissible values of the variable, but not while the variable remains a 
variable, whose value is not assigned ; and although they may, for every 
admissible value of the variable, imply the corresponding value of some 
other propositional function, yet while the variable remains as a variable 
they can imply nothing. The question concerning the nature of a 
propositional function as opposed to a proposition, and generally of a 
function as opposed to its values, is a difficult one, which can only be 
solved by an analysis of the nature of the variable. It is important, 
however, to observe that propositional functions, as was shown in 
Chapter vii, are more fundamental than other functions, or even than 
relations. For most purposes, it is convenient to identify the function 
and the relation, i.e.^ if y—f{x) is equivalent to xRy^ where R is a 
relation, it is convenient to speak of R as the function, and this will be 
done in what follows ; the reader, however, should remember that the 
idea of functionality is more fundamental than that of relation. But 
the investigation of these points has been already undertaken in Part I, 
and enough has been said to illustrate how a proposition may be a 
function of a variable. 

Other instances of non-numerical functions are afforded by diction- 
aries. The French for a word is a function of the English, and vice 
versd^ and both are functions of the term which both designate. The 
press-mark of a book in a library catalogue is a function of the book, 
and a number in a cipher is a function of the word for which it stands. 
In all these cases there is a relation by which the relatum becomes unique 
(or, in the case of languages, generally unique) when the referent 
is given ; but the terms of the independent variable do not form a 
series, except in the purely external order resulting from the alphabet. 

255. Let us now introduce the second specification, that our 
independent variable is to be a series. The dependent variable is then 
a series by correlation, and may be also an independent series. For 
example, the positions occupied by a material point at a series of instants 



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254, 255] The Correlatio7i of Series 265 

form a series by correlation with the instants, of which they are a 
function ; but in virtue of the continuity of motion, they also form, 
as a rule, a geometrical series independent of all reference to time. 
Thus motion affords an admirable example of the correlation of series. 
At the same time it illustrates a most important mark by which, when it 
is present, we can tell that a series is not independent. When the 
time is known, the position of a material particle is uniquely determined; 
but when the position is given, there may be several moments, or even an 
infinite number of them, corresponding to the given position. (There 
will be an infinite number of such moments if, as is commonly said, the 
particle has been at rest in the position in question. Rest is a loose and 
ambiguous expression, but I defer its consideration to Part VII.) Thus 
the relation of the time to the position is not strictly one-one, but may 
be many-one. This was a case considered in our general account of 
correlation, as giving rise to dependent series. We inferred, it will 
be remembered, that two correlated independent series are mathemati- 
cally on the same level, because if P, Q be their generating relations, and 
R the correlating relation, we infer F = RQR from Q = RFR. But 
this inference fails as soon as R is not strictly one-one, since then we no 
longer have RR contained in 1', where 1' means identity. For example, 
my father's son need not be myself, though my son's father must be. 
This illustrates the fact that, if i^ be a many-one relation, RR and RR 
must be carefully distinguished : the latter is contained in identity, but 
not the former. Hence whenever J? is a many-one relation, it may be 
used to form a series by correlation, but the series so formed cannot be 
independent. This is an important point, which is absolutely fatal to 
the relational theory of time*. For the present let us return to the 
case of motion. When a particle describes a closed curve, or one 
which has double points, or when the particle is sometimes at rest 
during a finite time, then the series of points which it occupies 
is essentially a series by correlation, not an independent series. But, 
as I remarked above, a curve is not only obtainable by motion, 
but is also a purely geometrical figure, which can be defined without 
reference to any supposed material point. When, however, a curve is 
so defined, it must not contain points of rest: the path of a material 
point which sometimes moves, but is sometimes at rest for a finite time, 
is different when considered kinematically and when considered geometri- 
cally; for geometrically the point in which there is rest is one, whereas 
kinematically it corresponds to many terms in the series. 

The above discussion of motion illustrates, in a non -numerical 
instance, a case which normally occurs among the functions of pure 
mathematics. These functions (when they are functions of a real 

* See my article '^'^Js position in Time and Space absolute or relative.^" Mind, 
July 1901. 



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266 Infinity and Continuity [chap, xxxil 

variable) usually fulfil the following conditions : Both the independent 
and the dependent variable are classes of numbers, and the defining 
relation of the function is many -one*. This case covers rational 
functions, circular and elliptic functions of a real variable, and the 
great majority of the direct functions of pure mathematics. In all such 
cases, the independent variable is a series of numbers, which may be 
restricted in any way we please — to positive numbers, rationals, integers, 
primes, or any other class. The dependent variable consists also of 
numbers, but the order of these numbers is determined by their relation 
to the corresponding term of the independent variable, not by that of 
the numbers forming the dependent variable themselves. In a large 
class of functions the two orders happen to coincide ; in others, again, 
where there are maxima and minima at finite intervals, the two orders 
coincide throughout a finite stretch, then they become exactly opposite 
throughout another finite stretch, and so on. If x be the independent 
variable, y the dependent variable, and the constitutive relation be 
many-one, the same number y will, in general, be a function of, i.e. 
correspond to, several numbers x. Hence the ^/-series is essentially by 
correlation, and cannot be taken as an independent series. If, then, we 
wish to consider the inverse function, which is defined by the converse 
relation, we need certain devices if we are still to have correlation of 
series. One of these, which seems the most important, consists in 
dividing the values of oo corresponding to the same value of y into 
classes, so that (what may happen) we can distinguish (say) n different 
og\^ each of which has a distinct one-one relation to «/, and is therefore 
simply reversible. This is the usual course, for example, in distinguish- 
ing positive and negative square roots. It is possible wherever the 
generating relation of our original function is formally capable of 
exhibition as a disjunction of one-one relations. It is plain that the 
disjunctive relation formed of n one-one relations, each of which contains 
in its domain a certain class u^ will, throughout the class u^ be an 
n-one relation. Thus it may happen that the independent variable 
can be divided into 7i classes, within each of which the defining relation 
is one-one, i.e. within each of which there is only one oo having the 
defining relation to a given y. In such cases, which are usual in pure 
mathematics, our many-one relation can be made into a disjunction of 
one-one relations, each of which separately is reversible. In the case of 
complex functions, this is, mutatis mutandis^ the method of Riemann 
surfaces. But it must be clearly remembered that, where our function 
is not naturally one-one, the y which appears as dependent variable is 
ordinally distinct from the y which appears as independent variable in 
the inverse function. 

The above remarks, which will receive illustration as we proceed, 

■^ I omit for the present complex variables^ which_, by introducing dimensions^ 
lead to complications of an entirely distinct kind. 



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255, 256] The Correlation oj Series 267 

have shown, I hope, how intimately the correlation of series is associated 
with the usual mathematical employment of functions. Many other 
cases of the importance of correlation will meet us as we proceed. It 
may be observed that every denumerable class is related by a one- valued 
function to the finite integers, and vice versa. As ordered by correlation 
with the integers, such a class becomes a series having the type of order 
which Cantor calls o). The fundamental importance of correlation to 
Cantor's theory of transfinite numbers will appear when we come to the 
definition of the transfinite ordinals. 

256. In connection with functions, it seems desirable to say some- 
thing concerning the necessity of a formula for definition. A function 
was originally, after it had ceased to be merely a power, essentially 
something that could be expressed by a formula. It was usual to start 
with some expression containing a variable x^ and to say nothing to 
begin with as to what oc was to be, beyond a usually tacit assumption 
that X was some kind of number. Any further limitations upon x were 
derived, if at all, from the formula itself; and it was mainly the desire 
to remove such limitations which led to the various generalizations of 
number. This algebraical generalization* has now been superseded by 
a more ordinal treatment, in which all classes of numbers are defined by 
means of the integers, and formulae are not relevant to the process. 
Nevertheless, for the use of functions, where both the independent and 
the dependent variables are infinite classes, the formula has a certain 
importance. Let us see what is its definition. 

A formula, in its most general sense, is a proposition, or more 
properly a prepositional function, containing one or more variables, 
a variable being any term of some defined class, or even any term 
without restriction. The kind of formula which is relevant in connection 
with functions of a single variable is a formula containing two variables. 
If both variables are defined, say one as belonging to the class u^ 
the other as belonging to the class z', the formula is true or false. It is 
true if every u has to every v the relation expressed by the formula ; 
otherwise it is false. But if one of the variables, say ^, be defined as 
belonging to the class w, while the other, ?/, is only defined by the 
formula, then the formula may be regarded as defining y as a function 
of w. Let us call the formula P^y. If in the class u there are terms x 
such that there is no term y which makes P^^/ ^ ^^^^ proposition, then 
the formula, as regards those terms, is impossible. We must therefore 
assume that u is a class every term of which will, for a suitable value 
of «/, make the proposition P^y true. If, then, for every term x of u^ 
there are some entities «/, which make P^y true, and others which do not 
do so, then P^^y correlates to every x a certain class of terms y. In 
this way y is defined as a function of x. 

* Of which an excellent account will be found in Couturat, De Vlnfini Mathema- 
tique, Paris^ 1896^ Part I, Book II. 



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268 Infinity and Continuity [chap, xxxil 

But the usual meaning oi formula in mathematics involves another 
element, which may also be expressed by the word law. It is difficult to 
say precisely what this element is, but it seems to consist in a certain 
degree of intensional simplicity of the proposition P^y. In the case of 
two languages, for example, it would be said that there is no formula 
connecting them, except in such cases as Grimm'^s law. Apart from the 
dictionary, the relation which correlates words in different languages is 
sameness of meaning ; but this gives no method by which, given a word 
in one language, we can infer the corresponding word in the other. 
What is absent is the possibility of calculation. A formula, on the 
other hand (say y — ^x)^ gives the means, when we know ^, of dis- 
covering y. In the case of languages, only enumeration of all pairs 
will define the dependent variable. In the case of an algebraical 
formula, the independent variable and the relation enable us to know 
all about the dependent variable. If functions are to extend to infinite 
classes, this state of things is essential, for enumeration has become 
impossible. It is therefore essential to the correlation of infinite classes, 
and to the study of functions of infinite classes, that the formula P^cy 
should be one in which, given x^ the class of terms y satisfying the 
formula should be one which we can discover. I am unable to give 
a logical account of this condition, and I suspect it of being purely 
psychological. Its practical importance is great, but its theoretical 
importance seems highly doubtful. 

There is, however, a logical condition connected with the above, 
though perhaps not quite identical with it. Given any two terms, 
there is some relation which holds between those two terms and 
no others. It follows that, given any two classes of terms u^ v^ 
there is a disjunctive relation which any one term of u has to at 
least one term of t^, and which no term not belonging to u has 
to any term. By this method, when two classes are both finite, 
we can carry out a correlation (which may be one-one, many-one, or 
one-many) which correlates terms of these classes and no others. In 
this way any set of terms is theoretically a function of any other; and 
it is only thus, for example, that diplomatic ciphers are made up. But 
if the number of terms in the class constituting the independent variable 
be infinite, we cannot in this way practically define a function, unless 
the disjunctive relation consists of relations developed one from the 
other by a law, in which case the formula is merely transferred to the 
relation. This amounts to saying that the defining relation of a function 
must not be infinitely complex, or, if it be so, must be itself a function 
defined by some relation of finite complexity. This condition, though 
it is itself logical, has again, I think, only psychological necessity, in 
virtue of which we can only master the infinite by means of a law of 
order. The discussion of this point, however, would involve a discussion 
of the relation of infinity to order — a question which will be resumed 



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256, 257] The Correlation of Series 269 

later, but which we are not yet in a position to treat intelligently. In 
any case, we may say that a formula containing two variables and 
defining a function must, if it is to be practically useful, give a relation 
between the two variables by which, when one of them is given, all the 
corresponding values of the other can be found ; and this seems to 
constitute the mathematical essence of all formulae. 

257. There remains an entirely distinct logical notion of much 
importance in connection with limits, namely the notion of a com- 
plete series. If R be the defining relation of a series, the series 
is complete when there is a term oo belonging to the series, such 
that every other term which has to x either the relation R or the 
relation R belongs to the series. It is connected (as was explained in 
Part IV) when no other terms belong to the series. Thus a complete 
series consists of those terms, and only those terms, which have the 
generating relation or its converse to some one term, together with that 
one term. Since the generating relation is transitive, a series which 
fulfils this condition for one of its terms fulfils it for all of them. 
A series which is connected but not complete will be called incomplete 
or partial. Instances of complete series are the cardinal integers, the 
positive and negative integers and zero, the rational numbers, the 
moments of time, or the points on a straight line. Any selection from 
such a series is incomplete with respect to the generating relations of the 
above complete series. Thus the positive numbers are an incomplete 
series, and so are the rationals between and 1. When a series is 
complete, no term can come before or after any term of the series 
without belonging to the series ; when the series is incomplete, this is 
no longer the case. A series may be complete with respect to one 
generating relation, but not with respect to another. Thus the finite 
integers are a complete series when the series is defined by powers of 
the relation of consecutiveness, as in the discussion of progressions in 
Part IV ; but when they are ordered by correlation with whole and part, 
they form only part of the series of finite and transfinite integers, as we 
shall see hereafter. A complete series may be regarded as the extension 
of a term with respect to a given relation, together with this term itself; 
and owing to this fact it has, as we shall find, some important differences 
from ordinally similar incomplete series. But it can be shown, by the 
Logic of Relations, that any incomplete series can be rendered complete 
by a change in the generating relation, and vice versa. The distinction 
between complete and incomplete series is, therefore, essentially relative 
to a given generating relation. 



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CHAPTER XXXIIL 



REAL NUMBERS. 

258. The philosopher may be surprised, after all that has already 
been said concerning numbers, to find that he is only now to learn about 
real numbers ; and his surprise will be turned to horror when he learns 
that real is opposed to rational. But he will be relieved to learn that 
real numbers are really not numbers at all, but something quite different. 
. -The series of real numbers, as ordinarily defined, consists of the 
whole assemblage of rational and irrational numbers, the irrationals 
being defined as the limits of such series of rationals as have neither 
a rational nor an infinite limit. This definition, however, introduces 
grave difficulties, which will be considered in the next chapter. For my 
part I see no reason whatever to suppose that there are any irrational 
numbers in the above sense ; and if there are any, it seems certain that 
they cannot be greater or less than rational numbers. When mathema- 
ticians have effected a generalization of number they are apt to be unduly 
modest about it — they think that the difference between the generalized 
and the original notions is less than it really is. We have already seen that 
the finite cardinals are not to be identified with the positive integers, nor 
yet with the ratios of the natural numbers to 1, both of which express 
relations, which the natural numbers do not. In like manner there is a 
real number associated with every rational number, but distinct from it. 
A real number, so I shall contend, is nothing but a certain class of 
rational numbers. Thus the class of rationals less than |^ is a real 
number, associated with, but obviously not identical with,- the rational 
number |. This theory is not, so far as I know, explicitly advocated by 
any other author, though Peano suggests it, and Cantor comes very near 
to it*. My grounds in favour of this opinion are, first, that such classes 
of rationals have all the mathematical properties commonly assigned 
to real numbers, secondly, that the opposite theory presents logical 
difficulties which appear to me insuperable. The second point will be 
discussed in the next chapter ; for the present I shall merely expound 

■^ Cf. Cantor^ Math. Annalen, Vol. xlvi^ § 10; Peano^ Rivista di Matematica, 
Vol. VI, pp. 126-140, esp. p. 133. 



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258, 259] Real Numbers 271 

my own view, and endeavour to show that real numbers, so understood, 
have all the requisite characteristics. It will be observed that the 
following theory is independent of the doctrine of limits, which will only 
be introduced in the next chapter. 

259. The rational numbers in order of magnitude form a series in 
which there is a term between any two. Such series, which in Part III 
we provisionally called continuous, must now receive another name, since 
we shall have to reserve the word continuous for the sense which Cantor 
has given to it. I propose to call such series compact^. The rational 
numbers, then, form a compact series. It is to be observed that, in a 
compact series, there are an infinite number of terms between any two, 
there are no consecutive terms, and the stretch between any two terms 
(whether these be included or not) is again a compact series. If now w^e 
consider any one rational number f, say r, we can define, by relation to r, 
four infinite classes of rationals : (1) those less than r, (2) those not 
greater than r, (3) those greater than r, (4) those not less than r. 
(2) and (4) differ from (1) and (B) respectively solely by the fact that 
the former contain r, while the latter do not. But this fact leads to 
curious differences of properties. (2) has a last term, while (1) has 
none; (1) is identical with the class of rational numbers less than a 
variable term of (1), while (2) does not have this characteristic. Similar 
remarks apply to (3) and (4), but these two classes have less importance 
in the present case than in (1) and (2). Classes of rationals having 
the properties of (1) are called segments. A segment of rationals mayj 
be_defijied a^s_ajcla^_of^^ nor yet coextensive^ 

with the rationals themselves {i.e. which contains some but not all 
rationals), and which is identical with the class of rationals less than a 
(variable) term of itself, i.e. with the class of rationals x such that there 
is a rational y of the said class such that x is less than y\. Now we shall 
find that segments are obtained by the above method, not only from 
single rationals, but also from finite or infinite classes of rationals, with 
the proviso, for infinite classes, that there must be some rational greater 
than any member of the class. This is very simply, done as follows. 

Let u be any finite or infinite class of rationals. Then four classes 
may be defined by relation to u% namely (1) those less than every u^ 
(2) those less than a variable u^ (B) those greater than every u^ (4) those 
greater than a variable ^^, i.e. those such that for each a term of u can be 
found which is smaller than it. If u be a finite class, it must have a maximum 
and a minimum term ; in this case the former alone is relevant to (2) 
and (3),^ the latter alotie to (1) and (4). Thus this case is reduced to 
the former, in which we had only a single rational. I shall therefore 

* Such series are called by Cantor ilherall dicht. 

t I shall for simplicity confine myself entirely to rationals without sign. The 
extension to such as are positive or negative presents no difficulty whatever. 
X See Formulaire de Mathematiques , Vol. ii^ Part in, § 61 (Turin^ 1899). 
§ Eight classes may be defined^ but four are all that we need. 



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272 Infinity and Continuity [chap, xxxiii 

assume in future that ti is an infinite class, and further, to prevent 
reduction to our former case, I shall assume, in considering (2) and (3), 
that u has no maximum, that is, that every term of u is less than some 
other term of u ; and in considering (1) and (4), I shall assume that u 
has no minimum. For the present I confine myself to (2) and (3), and 
I assume, in addition to the absence of a maximum, the existence of 
rationals greater than any u^ that is, the existence of the class (3). 
Under these circumstances, the class (S) will be a segment. For (S) 
consists of all rationals which are less than a variable u ; hence, in the 
first place, since u has no maximum, (S) contains the whole of w. In the 
second place, since every term of {%) is less than some u^ which in turn 
belongs to (2), every term of (2) is less than some other term of (^) ; 
and every term less than some term of (S) is a fortiori less than some u^ 
and is therefore a term of (£). Hence (^) is identical with the class of 
terms less than some term of (2), and is therefore a segment. 

Thus we have the following conclusion : If u be a single rational, or 
a class of rationals all of which are less than some fixed rational, then 
the rationals less than u^ if u be a single term, or less than a variable 
term of u^ if u be a class of terms, always form a segment of rationals. 
My contention is, that a segment of rationals is a real number. 

260. So far, the method employed has been one which may be 
employed in any compact series. In what follows, some of the theorems 
will depend upon the fact that the rationals are a denumerable series. 
I leave for the present the disentangling of the theorems dependent 
upon this fact, and proceed to the properties of segments of rationals. 

Somfi-Sfigments, as we have seen, consist of the rationals less than 
some given rational. Some, it will be found, though not so defined, are 
nevertheless capable of being so defined. For example, the rationals 
less than a variable term of the series '9, '99, '999, etc.^ are the same as 
the rationals less than 1. But other segments, which correspond to 
what are usually called irrationals, are incapable of any such definition. 
How this fact has led to irrationals we shall see in the next chapter. 
For the present I merely wish to point out the well-known fact that 
segments are not capable of a one-one correlation with rationals. There 
are classes of rationals defined as being composed of all terms less than 
a variable term of an infinite class of rationals, which are not definable 
as all the rationals less than some one definite rational*. Moreover 
there are more segments than rationals, and hence the series of segments 
has continuity of a higher order than the rationals. Segments form a 
series in virtue of the relation of whole and part, or of logical inclusion 
(excluding identity). Any two segments are such that one of them 
is wholly contained in the other, and in virtue of this fact they form 
a series. It can be easily shown that they form a compact series. 
What is more remarkable is this : if we apply the above process to the 

^ Cf. Part I, chap, v, p. 60. 



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259, 260] Beal Numbers 273 

series of segments, forming segments of segments by reference to 
classes of segments, we find that every segment of segments can be 
defined as all segments contained in a certain definite segment. Thus 
the segment of segments defined by a class of segments is always 
identical with the segment of segments defined by some one segment. 
Also every segment defines a segment of segments which can be defined 
by an infinite class of segments. These two properties render the 
series of segments perfect^ in Cantor's language ; but the explanation of 
this term must be left till we come to the doctrine of limits. 

We might have defined our segments as all rationals greater than 
some term of a class u of rationals. If we had done this, and inserted 
the conditions that it was to have no minimum, and that there were to 
be rationals less than every u^ we should have obtained what may be 
called upper segments, as distinguished from the former kind, which 
may be called lower segments. We should then have found that, corre- 
sponding to every upper segment, there is a lower segment which contains 
all rationals not contained in the upper segment, with the occasional 
exception of a single rational. There will be one rational not belonging 
to either the upper or the lower segment, when the upper segment 
can be defined as all rationals greater than a single rational. In this 
case, the corresponding lower segment will consist of all rationals less 
than this single rational, which will itself belong to neither segment. 
Since there is a rational between any two, the class of rationals not 
greater than a given rational cannot ever be identical with the class of 
rationals less than some other; and a class of rationals having a 
maximum can never be a segment. Hence it is impossible, in the case 
in question, to find a lower segment containing all the rationals not 
belonging to the given upper segment. But when the upper segment 
cannot be defined by a single rational, it will always be possible 
to find a lower segment containing all rationals not belonging to the 
upper segment. 

Zero and infinity may be introduced as limiting cases of segments, 
but in the case of zero the segment must be of the kind which we 
called (1) above, not of the kind (2) hitherto discussed. It is easy to 
construct a class of rationals such that some term of the class will be less 
than any given rational. In this case, the class (1) will contain no terms, 
and will be the null-class. This is the real number zero, which, however, 
is not a segment, since a segment was defined as a class which is not null. 
In order to introduce zero as a class of the kind which we called (2), we 
should have to start with a null class of rationals. No rational is less 
than a term of a null class of rationals, and thus the class (2), in such a 
case, is null. Similarly the real number infinity may be introduced. 
This is identical with the whole class of rationals. If we have any 
class It of rationals such that no rational is greater than all ^^'s, then 
every rational is contained in the class of rationals less than some 
R. 18 



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274 Infinity and Continuity [chap, xxxiii 

u. Or again, if we have a class of rationals of which a term is less than 
any assigned rational, the resulting class (4) (of terms greater than 
some u) will contain every rational, and will thus be the real number 
infinity. Thus both zero and infinity may be introduced as extreme 
terms among the real numbers, but neither is a segment according to the 
definition. 

261. A given segment may be defined by many different classes of 
rationals. Two such classes u and v may be regarded as having the 
segment as a common property. Two infinite classes u and v will define 
the same lower segment if, given any u^ there is a t; greater than it, and 
given any v^ there is a u greater than it. If each class has no maximum, 
this is also a necessary condition. The classes u and v are then what 
Cantor calls coherent {zusammengehorig). It can be shown, without 
considering segments, that the relation of being coherent is symmetrical 
and transitive*, whence we should infer, by the principle of abstraction, 
that both have to some third term a common relation which neither has 
to any other term. This third term, as we see from the preceding 
discussion, may be taken to be the segment which both define. We 
may extend the word coherent to two classes u and v^ of which one 
defines an upper segment, the other a lower segment, which between 
them include all rationals with at most one exception. Similar remarks, 
mutatis mutandis^ will still apply in this case. 

We have now seen that the usual properties of real numbers belong 
to segments of rationals. There is therefore no mathematical reason for 
distinguishing such segments from real numbers. It remains to set 
forth, first the nature of a limit, then the current theories of irrationals, 
and then the objections which make the above theory seem preferable. 

Note. The above theory is virtually contained in Professor Peano's 
article already referred to (" Sui Numeri Irrazionali,'' Rivista di Mate- 
matica, vi, pp. 126 — 140), and it was from this article, as well as from the 
Formidahr de Mathematiques^ that I was led to adopt the theory. In 
this article, separate definitions of real numbers (§ S, No. 5) and of 
segments (§ 8, '0) are given, which makes it seem as though the two 
were distinguished. But after the definition of segments, we find the 
remark (p. IBS) : " Segments so defined differ only in nomenclature from 
real numbers.'" Professor Peano proceeds first to give purely technical 
reasons for distinguishing the two by the notation, namely that the 
addition, subtraction, etc. of real numbers is to be differently conducted 
from analogous operations which are to be performed on segments. 
Hence it would appear that the whole of the view I have advocated is 
contained in this article. At the same time, there is some lack of 
clearness, since it appears from the definition of real numbers that they 
are regarded as the limits of classes of rationals, whereas a segment is 

* Cf. Cantor^ Math. Annaleii, xlvi^ and Rivista di Matematica, y, pp. 158^ 159. 



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260, 261] Real Numbers 275 

in no sense a limit of a class of rationals. Also it is nowhere suggested — 
indeed, from the definition of real numbers the opposite is to be 
inferred — that no real number can be a rational, and no rational can be 
a real number. And this appears where he points out (p. 134) that 1 
differs from the class of proper fractions (which is no longer the case as 
regards the reah number 1, when this is distinguished both from the 
integer 1 and from the rational number 1 : 1), or that we say 1 is less 
than V^ (in which case, I should say, 1 must be interpreted as the class 
of proper fractions, and the assertion must be taken to mean : the 
proper fractions are some, but not all, of the rationals whose square 
is less than %). And again he says {ib.) : " The real number, although 
determined by, and determining, a segment li^ is commonly regarded as 
the extremity, or end, or upper limit, of the segment"'; whereas there is 
no reason to suppose that segments not having a rational limit have a 
limit at all. Thus although he confesses {ib,) that a complete theory 
of irrationals can be constructed by means of segments, he does not 
seem to perceive the reasons (which will be given in the next chapter) 
why this must be done — reasons which, in fact, are rather philosophical 
than mathematical. 



18—2 



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CHAPTER XXXIV. 



LIMITS AND IRRATIONAL NUMBERS. 

262. The mathematical treatment of continuity rests wholly upon 
the doctrine of limits. It has been thought by some mathematicians 
and some philosophers that this doctrine had been superseded by the 
Infinitesimal Calculus, and that this has shown true infinitesimals 
to be presupposed in limits*. But modern mathematics has shown, 
conclusively as it seems to me, that such a view is erroneous. The 
method of limits has more and more emerged as fundamental. In this 
Chapter, I shall first set forth the general definition of a limit, and 
then examine its application to the creation of irrationals. 

A compact series we defined as one in w^hich there is a term between 
any two. But in such a series it is always possible to find two classes of 
terms which have no term between them, and it is always possible to 
reduce one of these classes to a single term. For example, if P be the 
generating relation and oo any term of the series, then the class of terms 
having to x the relation P is one between which and x there is no term-|". 
The class of terms so defined is one of the two segments determined 
by 00 ; the idea of a segment is one which demands only a series in 
general, not necessarily a numerical series. In this case, if the series be 
compact, X is said to be the limit of the class ; when there is such a 
term as ^, the segment is said to be terminated, and thus every 
terminated segment in a compact series has its defining term as a limit. 
But this does not constitute a definition of a limit. To obtain the 
general definition of a limit, consider any class u contained in the series 
generated by P. Then the class it will in general, with respect to any 
term x not belonging to it, be divisible into two classes, that whose 
terms have to x the relation P (which I shall call the class of terms pre- 
ceding x\ and that whose terms have to x the relation P (which I shall 
call the class of terms following x). If x be itself a term of u^ we 

* This is the view^ for instance^ of Colien_, Das Princip dej' Injinitesimal- 
Methode und seAne Geschichte, Berlin, 1883; see pp. 1, 2. 

t It is perhaps superfluous to explain that a term is between two classes u, v, when 
it has the relation F to every term of u, and the relation P to every term of Vy or 
vice versa. 



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262-264] Limits and Irrational Numbers 277 

consider all the terms of u other than x^ and these are still divisible into 
the above two classes, which we may call ituOO and ifuoo respectively. 
If, now, iTitOO be such that, if y be any term preceding ^, there is a term 
of itmOc following ?/, i.e. between oo and «/, then ^ is a limit of nvuoo. Similarly 
if TTuOC be such that, if z be any term after oo^ there is a term of tTuOO 
between x and ^, then <r is a limit of ttuOO. We now define that x is 
a limit of it if it is a limit of either ituOO or ^uX. It is to be observed that 
u may have many limits, and that all the limits together form a new 
class contained in the series generated by P. This is the class (or rather 
this, by the help of certain further assumptions, becomes the class) 
which Cantor designates as the first derivative of the class u, 

263. Before proceeding further, it may be well to make some 
general remarks of an elementary character on the subject of limits. 
In the first place, limits belong usually to classes contained in compact 
series — classes which may, as an extreme case, be identical with the 
compact series in question. In the second place, a limit may or may 
not belong to the class u of which it is a limit, but it always belongs to 
some series in which u is contained, and if it is a term of u^ it is still a 
limit of the class consisting of all terms of u except itself. In the 
third place, no class can have a limit unless it contains an infinite 
number of terms. For, to revert to our former division, if u be finite, 
iTuOO and ituX will both be finite. Hence each of them will have a term 
nearest to x^ and between this term and x no term of u will lie. Hence 
X is not a limit of u ; and since x is any term of the series, it will have 
no limits at all. It is common to add a theorem that every infinite 
class, provided its terms are all contained between two specified terms 
o^ the series generated by P, must have at least one limit ; but this 
theorem, we shall find, demands an interpretation in terms of segments, 
and is not true as it stands. In the fourth place, if u be co-extensive 
with the whole compact series generated by P, then every term of this 
series is a limit of u. There can be no other terms that are limits 
in the same sense, since limits have only been defined in relation to this 
compact series. To obtain other limits, we should have to regard the 
series generated by P as forming part of some other compact series — a 
case which, as we shall see, may arise. In any case, if u be any compact 
series, every term of z^ is a limit of u ; whether u has also other limits, 
depends upon further circumstances. A limit may be defined generally 
as a term which immediately follows (or precedes) some class of terms 
belonging to an infinite series, without immediately following (or 
preceding, as the case may be) any one term of the series. In this way, 
w^e shall find, limits may be defined generally in all infinite series which 
are jot progressions — as, for instance, in the series of finite and trans- 
finite integers. 

264. We may now proceed to the various arithmetical theories of 
irrationals, all of which depend upon limits. We shall find that, in the 



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278 Infinity and Continuity [chap, xxxiv 

exact form in which they have been given by their inventors, they all 
involve an axiom, for which there are no arguments, either of philo- 
sophical necessity or of mathematical convenience; to which there are 
grave logical objections; and of which the theory of real numbers given 
in the preceding Chapter is wholly independent. 

Arithmetical theories of irrationals could not be treated in Part II, 
since they depend essentially upon the notion of order. It is only by 
means of them that numbers become continuous in the sense now usual 
among mathematicians ; and we shall find in Part VI that no other 
sense of continuity is required for space and time. It is very important 
to realize the logical reasons for which an arithmetical theory of 
irrationals is imperatively necessary. In the past, the definition of 
irrationals was commonly effected by geometrical considerations. This 
procedure was, however, highly illogical ; for if the application of 
numbers to space is to yield anything but tautologies, the numbers 
applied must be independently defined ; and if none but a geometrical 
definition were possible, there would be, properly speaking, no such 
arithmetical entities as the definition pretended to define. The alge- 
braical definition, in which irrationals were introduced as the roots 
of algebraic equations having no rational roots, was liable to similar 
objections, since it remained to be shown that such equations have 
roots ; moreover this method will only yield the so-called algebraic 
numbers, which are an infinitesimal proportion of the real numbers, and 
do not have continuity in Cantor's sense, or in the sense required by 
Geometry. And in any case, if it is possible, without any further 
assumption, to pass from Arithmetic to Analysis, from rationals to 
irrationals, it is a logical advance to show how this can be done. 
The generalizations of number — with the exception of the intro- 
duction of imaginaries, which must be independently effected — are all 
necessary consequences of the admission that the natural numbers form 
a progression. In every progression the terms have two kinds of 
relations, the one constituting the general analogue of positive and 
negative integers, the other that of rational numbers. The rational 
numbers form a denumerable compact series ; and segments of a denumer- 
able compact series, as we saw in the preceding Chapter, form a series 
which is continuous in the strictest sense. Thus all follo¥/s from the 
assumption of a progression. But in the present Chapter we have to 
examine irrationals as based on limits ; and in this sense, we shall find 
that they do not follow without a new assumption. 

There are several somewhat similar theories of irrational numbers. 
I will begin with that of Dedekind*. 

265. Although rational numbers are such that, between any two, 
there is always a third, yet there are many ways of dividing all rational 

* Sietigkeit und irrationale Zahlen, 2nd ed., Brunswick^ 1892. 



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264-266] Limits and Irrational Numbers 279 

numbers into two classes, such that all numbers of one class come after 
all numbers of the other class, and no rational number lies between the 
two classes, while yet the first class has no first term and the second has 
no last term. For example, all rational numbers, without exception, 
may be classified according as their squares are greater or less than 2, 
All the terms of both classes may be arranged in a single series, in which 
there exists a definite section, before which comes one of the classes, 
and after which comes the other. Continuity seems to demand that 
some term should correspond to this section. A number which lies 
between the tw^o classes must be a new number, since all the old numbers 
are classified. This new number, which is thus defined by its position in 
a series, is an irrational number. When these numbers are introduced, 
not only is there always a number between any two numbers, but there 
is a number between any two classes of which one comes wholly after the 
other, and the first has no minimum, while the second has no maximum. 
Thus we can extend to numbers the axiom by which Dedekind defines 
the continuity of the straight line {op, cit. p. 11) : — 

" If all the points of a line can be divided into two classes such that 
every point of one class is to the left of every point of the other class, 
then there exists one and only one point which brings about this 
division of all points into two classes, this section of the line into 
two parts. ^' 

266. This axiom of Dedekind's is, however, rather loosely worded, and 
requires an emendation suggested by the derivation of irrational numbers. 
If all the points of a line are divided into two classes, no point is left 
over to represent the section. If all be meant to exclude the point repre- 
senting the section, the axiom no longer characterizes continuous series, 
but applies equally to all series, e.g. the series of integers. The axiom 
must be held to apply, as regards the division, not to all the points of the 
line, but to all the points forming some compact series, and distributed 
throughout the line, but consisting only of a portion of the points 
of the line. When this emendation is made, the axiom becomes ad- 
missible. If, from among the terms of a series, some can be chosen 
out to form a compact series which is distributed throughout the 
previous series ; and if this new series can always be divided in 
Dedekind's manner into two portions, between which lies no term of 
the new series, but one and only one term of the original series, then 
the original series is continuous in Dedekind's sense of the word. The 
emendation, however, destroys entirely the self-evidence upon which 
alone Dedekind relies (p. 11) for the proof of his axiom as applied 
to the straight line. 

Another somewhat less complicated emendation may be made, which 
gives, I think, what Dedekind meant to state in his axiom. A series, 
we may say, is continuous in Dedekind^s sense when, and only when, 
if all the terms of the series, without exception, be divided into two 



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280 Infinity and Continuity [chap, xxxiv 

classes, su(^h that the whole of the first class precedes the whole of 
the second, then, however the division be effected, either the first class 
has a last term, or the second class has a first term, but never both. 
This term, which comes at one end of one of the two classes, may then 
be used, in Dedekind's manner, to define the section. In discrete series, 
such as that of finite integers, there is both a last term of the first 
class and a first term of the second class * ; while in compact series 
such as the rationals, where there is not continuity, it sometimes, 
happens (though not for every possible division) that the first class 
has no last term and the last class has no first term. Both these cases 
are excluded by the above axiom. . But I cannot see any vestige of 
self-evidence in such an axiom, either as applied to numbers or as applied 
to space. 

267. Leaving aside, for the moment, the general problem of con- 
tinuity, let us return to Dedekind's definition of irrational numbers. 
The first question that arises is this : What right have we to assume 
the existence of such numbers ? What reason have we for supposing 
that there must be a position between two classes of which one is wholly 
to the right of the other, and of which one has no minimum and the 
other no maximum ? This is not true of series in general, since many 
series are discrete. It is not demanded by the nature of order. And, 
as we have seen, continuity in a certain sense is possible without it. 
Why then should we postulate such a number at all ? It must be 
remembered that the algebraical and geometrical problems, which ir- 
rationals are. intended to solve, must not here be brought into the 
account. The existence of irrationals has, in the past, been inferred 
from such problems. The equation oc^ — ^ — must have a root, it was 
argued, because, as w grows from to S, ^^ — 2 increases, and is first 
negative and then positive ; if w changes continuously, so does ^ - 2 ; 
hence oo^—^ must assume the value in passing from negative to positive. 
Or again, it was argued that the diagonal of unit square has evidently si- 
precise and definite length oc^ and that this length is such that ^^ — 2 = 0. 
But such arguments were powerless to show that x is truly a number. 
They might equally well be regarded as showing the inadequacy of 
numbers to Algebra and Geometry. The present theory is designed 
to prove the arithmetical existence of irrationals. In its design, it is 
preferable to the previous theories ; but the execution seems to fall short 
of the design. 

Let us examine in detail the definition of J^ by Dedekind's method. 
It is a singular fact that, although a rational number lies between any 
two single rational numbers, two classes of rational numbers may be 
defined so that no rational number lies between them, though all ot 

■^ If the series contains a proper part which is a progression, it is only true in 
general, not without exception, that the first class must have a last term. 



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266, 267] Limits and Irrational Numbers 281 

one class are higher than all of the other. It is evident that one at 
least of these classes must consist of an infinite number of terms. For 
if not, we could pick out the two of opposite kinds which were nearest 
together, and insert a new number between them. This one would be 
between the two classes, contrary to the hypothesis. But when one of 
the classes is infinite, we may arrange all or some of the terms in a series 
of terms continually approaching the other class, without reaching it, 
and without having a last term. Let us, for the moment, suppose our 
infinite class to be denumerable. We then obtain a denumerable series 
of numbers a^, all belonging to the one class, but continually approaching 
the other class. Let ^ be a fixed number of the other class. Then 
between an and B there is always another rational number; but this 
may be chosen to be another of the a\^ say a^+i ; and since the series of 
d^ is infinite, we do not necessarily obtain, in this way, any number not 
belonging to the series of a\. In the definition of irrationals, the series of 
F^ is also infinite. Moreover, if the &'s also be denumerable, any rational 
number between a^ and ^^, for suitable values of p and q^ either is an^p 
or &m + g? or else lies between a^j^^ and <2^_up+i or between h^ + q^^^ hm+q+i- 
In fact, a^+p always lies between a^ and h^- I^J successive steps, no term 
is obtained which lies between all the 6's and all the a's. Nevertheless, 
both the <^'s and the F^ are convergent. For, let the a^ increase, while 
the 6's diminish. Then bn — ctn and bn — an+i continually diminish, and 
therefore an^i — an^ which is less than either, is less than a continually 
diminishing number. Moreover this number diminishes without limit ; 
for if bn — an had a hmit e, the number an + e/2 would finally lie between 
the two classes. Hence an + i — ccn becomes finally less than any assigned 
number. Thus the a's and b's are both convergent. Since, moreover, 
their difference may be made less than any assigned number 6, they have 
the same limit, if they have any. But this limit cannot be a rational 
number, since it lies between all the ^'s and all the ^'s. Such seems 
to be the argument for the existence of irrationals. For ex:ample, if 

Thus ^ = S + 1/^ = 2 4- ^ , and ^-1 = 1 +-^rT^r7 ~ = ^^^* 

111 

The successive convergents to the continued fraction 1 + ^-~- - ^ „-— - 

are such that all the odd convergents are less than all the even con- 
vergents, while the odd convergents continually grow, and the even 
ones continually diminish. Moreover the difference between the odd 
and the next even convergent continually diminishes. Thus both 
series, if they have a limit, have the same limit, and this limit is 
defined as \/2. 

But the existence of a limit, in this case, is evidently a sheer as- 
sumption. In the beginning of this Chapter, we saw that the existence 



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282 Infinity and Continuity [chap, xxxiv 

of a limit demands a larger series of which the limit forms part. To 
create the limit by means of the series whose limit is to be found would 
therefore be a logical error. It is essential that the distance from the 
limit should diminish indefinitely. But here, it is only the distance of 
consecutive terms which is known to diminish indefinitely. Moreover 
all the ^'s are less than 5^. Hence they continually differ less and less 
from hn- But whatever n may be, bn cannot be the limit of the «'s, 
for hn-^\ lies between h^ and all the ^'s. This cannot prove that a limit 
exists, but only that, if'it existed, it would not be any one of the «'s or 
6's, nor yet any other rational number. Thus irrationals are not proved 
to exist, but may be merely convenient fictions to describe the relations 
of the cC^ and 6'*s. 

268. The theory of Weierstrass concerning irrationals is somewhat 
similar to that of Dedekind. In Weierstrass's theory, we have a series 
of terms ^i, 6^2, . . , <^n? • • ? such that S a^^ for all values of n^ is less 
than some given number. This case is presented, e.g.^ by an infinite 
decimal. The fraction 3'14159 . . . , however many terms we take, 
remains less than 3*1416. In this method, as Cantor points out*, the 
limit is riot created by the summation, but must be supposed to exist 

already in order that S a^ may be defined by means of it. This is the 

1 
same state of things as we found in Dedekind's theory : series of rational 
numbers cannot prove the existence of irrational numbers as their limits, 
but can only prove that, i/" there is a limit, it must be irrational. 

Thus the arithmetical theory of irrationals, in either of the above 
forms, is liable to the following objections. (1) No proof is obtained 
from it of the existence of other than rational numbers, unless we 
accept some axiom of continuity different from that satisfied by 
rational numbers ; and for such an axiom we have as yet seen no 
ground. (^) Granting the existence of irrationals, they are merely 
specified, not defined, by the series of rational numbers whose limits 
they are. Unless they are independently postulated, the series in 
question cannot be known to have a limit ; and a knowledge of the 
irrational number which is a limit is presupposed in the proof that 
it is a limit. Thus, although without any appeal to Geometry, any 
given irrational number can be specified by means of an infinite series 
of rational numbers, yet, from rational numbers alone, no proof can 
be obtained that there are irrational numbers at all, and their existence 
must be proved from a new and independent postulate. 

Another objection to the above theory is that it supposes rationals 
and irrationals to form part of one and the same series generated by 
relations of greater and less. This raises the same kind of difficulties as 
we found to result, in Part II, from the notion that integers are greater 

■^ Mannichfaltigkeitslehre, p. 22. I quote Weierstrass's theory from the account 
in Stolz, Vorlesungen uber aUgemeine Arithmetik, i. 



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267-269] Limits and Irrational Numbers 283 

or less than rationals, or that some rationals are integers. Rationals 
are essentially relations between integers, but irrationals are not such 
relations. Given an infinite series of rationals, there may be two 
integers whose relation is a rational which limits the series, or there 
may be no such pair of integers. The entity postulated as the limit, 
in this latter case, is no longer of the same kind as the terms of 
the series which it is supposed to limit ; for each of them is, while 
the limit is not, a relation between two integers. Of such hetero- 
geneous terms, it is difficult to suppose that they can have relations 
of greater and less ; and in fact, the constitutive relation of greater 
and less, from which the series of rationals springs, has to receive 
a new definition for the case of two irrationals, or of a rational and 
an irrational. This definition is, that an irrational is greater than a 
rational, when the irrational limits a series containing terms greater 
than the given rational. But what is really given here is a relation 
of the given rational to a class of rationals, namely the relation of 
belonging to the segment defined by the series whose limit is the given 
irrational. And in the case of two irrationals, one is defined to be 
greater than the other when its defining series contains terms greater 
than any terms of the defining series of the other — a condition which 
amounts to saying that the segment corresponding to the one contains 
as a proper part the segment corresponding to the other. These 
definitions define a relation quite different from the inequality of two 
rationals, namely the logical relation of inclusion. Thus the irrationals 
cannot form part of the series of rationals, but new terms corresponding 
to the rationals must be found before a single series can be constructed. 
Such terms, as we saw in the last chapter, are found in segments ; but 
the theories of Dedekind and Weierstrass leave them still to seek. 

269. The theory of Cantor, though not expressed, philosophically 
speaking, with all the requisite clearness, lends itself more easily to the 
interpretation which I advocate, and is specially designed to prove 
the existence of limits. He remarks* that, in his theory, the existence 
of the limit is a strictly demonstrable proposition ; and he strongly 
emphasizes the logical error involved in attempting to deduce the 
existence of the limit from the series whose limit it is (i5., p. 2S)+. 
Cantor starts by considering what he calls fundamental series (which 
are the same as what I have called progressions) contained in a larger 
series. Each of these fundamental series is to be wholly ascending or 
wholly descending. Two such series are called coherent {zusammenge- 
hm^ig) under the following circumstances : — 

^ Op. cit., p. 24. 

t Cantor's theory of irrationals will be found in op. cit., p. 23^ and in Stolz, 
Vorlesungen uber allgemeine Arithmetik, i, 7- I shall follow, to begin with, a later 
account, which seems to me clearer ; this forms § 10 in an article contained in Math. 
Annalen, xlvi, and in Rivista di Matematica, v. 



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284 Infinity and Continuity [chap. xxxiy 

(1) If both are ascending, and after any term of either there is 
always a term of the other ; 

(2) If both are descending, and before any term of either there is 
always a term of the other ; 

(3) If one is ascending, the other descending, and the one wholly 
precedes the other, and there is at most one term which is between the 
two fundamental series. 

The relation of being coherent is symmetrical, in virtue of the 
definition ; and Cantor shows that it is transitive. In the article from 
which the above remarks are extracted. Cantor is dealing with more 
general topics than the definition of irrationals. But the above general 
account of coherent series will help us to understand the theory of 
irrationals. This theory is set forth as follows in the Manmchfaltig- 
'keitslehre (p. S3 K) : — 

A fundamental series of rationals is defined as a denumerable series 
such that, given any number e, there are at most a finite number of 
terms in the series the absolute values of whose differences from sub- 
sequent terms exceed 6. That is to say, given any number e, however 
small, any tw^o terms of the series which both come after a certain term 
have a difference which lies between + e and — e. Such series must be 
of one of three kinds : (1) Any number e being mentioned, the absolute 
values of the terms, from some term onwards, will all be less than e, 
whatever e may be ; (2) from some term onwards, all the terms may be 
greater than a certain positive number p ; (3) from some term onwards, 
all the terms may be less than a certain negative number —p. A real 
number h is to be defined by the fundamental series, and is said in the 
first case to be zero, in the second to be positive, and in the third to 
be negative. To define the addition, etc.^ of these new real numbers, 
we observe that, if a^,^ aj be the z^th terms of two fundamental series, 
the series whose z^th term is a^ -\- a^ or a^, — a J or a^, x a^ is also a funda- 
mental series ; while if the real number defined by the series {a^) * is 
not zero, (aj / a^,) also defines a fundamental series. If 6, 6' be the real 
numbers defined by the series (a^), (a/), the real numbers defined by 
(a^ 4- <3^/), (a^, — aj)^ {a^, x aj) and {aj j a^) are defined to be Z^ + 6", 5 — h\ 
h xh' and h' j h respectively. Hence we proceed to the definitions of equal, 
greater and less among real numbers. We define that h=b' means b — b'=='0; 
h>h' means that h — h' is positive ; and b <h' means that b — b' is 
negative — all terms which have been already defined. Cantor remarks 
further that in these definitions one of the numbers may be rational. 
This may be formally justified, in part, by the remark that a denu- 
merable series whose terms are all one and the same rational number is 
a fundamental series, according to the definition ; hence in constructing 

"^ The symbol {av) denotes the whole series whose i/th term is av, not this term 
alone. 



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269, 270] Limits and Irrational Numbers 285 

the differences a^, — aj^ by which 6 — h' is defined, we may put some fixed 
rational a in place of aj for all values of v. But the consequence that 
we can define h — a does not follow, and that for the following reason. 
There is absolutely nothing in the above definition of the real numbers 
to show that a is the real number defined by a fundamental series whose 
terms are all equal to a. The only reason why this seems self-evident is, 
that the definition by limits is unconsciously present, making us think 
that, since a is plainly the limit of a series whose terms are all equal 
to a^ therefore <2 must be the real number defined by such a series. 
Since, however, Cantor insists — rightly, as I think — that his method 
is independent of limits, which, on the contrary, are to be deduced from 
it (pp. ^4 — 5), we must not allow this preconception to weigh with us. 
And the preconception, if I am not mistaken, is in fact erroneous. 
There is nothing in the definitions above enumerated to show that a 
real number and a rational number can ever be either equal or unequal, 
and there are very strong reasons for supposing the contrary. Hence 
also we must reject the proposition (p. 24) that, if h be the real number 
defined by a fundamental series (a^), then 

Lim a^, — h. 

Cantor is proud of the supposed fact that his theory renders this pro- 
position strictly demonstrable. But, as we have seen, there is nothing 
to show that a rational can be subtracted from a real number, and 
hence the supposed proof is fallacious. What is true, and what has 
all the mathematical advantages of the above theorem, is this : Con- 
nected with every rational a there is a real number, namely that defined 
by the fundamental series whose terms are all equal to a\ if 6 be the 
real number defined by a fundamental series {a^) and if h^ be the real 
number defined by a fundamental series whose terms are all equal to <2^, 
then (6^) is a fundamental series of real numbers whose limit is h. But 
from this we cannot infer, as Cantor supposes (p. 24), that Lim a^, exists ; 
this will only be true in the case where {a^) has a rational limit. The 
limit of a series of rationals either does not exist, or is rational ; in no 
case is it a real number. But in all cases a fundamental series of 
rationals defines a real number, which is never identical with any 
rational. 

270. Thus to sum up what has been said on Cantor's theory : By 
proving that two fundamental series may have the relation of being 
coherent, and that this relation is symmetrical and transitive. Cantor 
shows, by the help of the principle of abstraction (which is tacitly 
assumed), that two such series both have some one relation to one third 
term, and to no other. This term, when our series consist of rationals, 
we define as the real number which both determine. We can then define 
the rules of operation for real numbers, and the relations of equal, 
greater and less between them. But the principle of abstraction leaves 



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286 Infinity and Continuity [chap, xxxiv 

us in doubt as to what the real numbers really are. One thing, however, 
seems certain. They cannot form part of any series containing rationals, 
for the rationals are relations between integers, while the real numbers 
are not so ; and the constitutive relation in virtue of which rationals 
form a series is defined solely by means of the integers between which 
they are relations, so that the same relation cannot hold between two 
real numbers, or between a real and a rational number. In this doubt 
as to what real numbers may be, we find that segments of rationals, as 
defined in the preceding chapter, fulfil all the requirements laid down 
in Cantor's definition, and also those derived from the principle of 
abstraction. Hence there is no logical ground for distinguishing seg- 
ments of rationals from real numbers. If they are to be distinguished, 
it must be in virtue of some immediate intuition, or of some wholly new 
axiom, such as, that all series of rationals must have a limit. But this 
would be fatal to the uniform development of Arithmetic and Analysis 
from the five premisses which Peano has found sufficient, and would be 
wholly contrary to the spirit of those who have invented the arithmetical 
theory of irrationals. The above thfeory, on the contrary, requires no new 
axiom, for if there are rationals, there must be segments of rationals ; 
and it removes what seems, mathematically, a wholly unnecessary 
complication, since, if segments will do all that is required of irrationals, 
it seems superfluous to introduce a new parallel series with precisely the 
same mathematical properties. I jconclude, then, that an irrational 
actually is a segment of rationals which does not have §^^^.^^h wKile 

a real number which would be commonly identified with a rational is a 
segment which does have a rational limit; and this applies, e.g..^ to the 
real number defined by a fundamental series of rationals whose terms 
are all equal. This is the theory which was set forth positively in the 
preceding Chapter, and to which, after examining the current theories of 
irrationals, we are again brought back. The greater part of it applies to 
compact series in general ; but some of the uses of fundamental series, 
as we shall see hereafter, presuppose either numerical measurement of 
distances or stretches, or that a denumerable compact series is contained 
in our series in a certain manner*. The whole of it, however, applies to 
any compact series obtained from a progression as the rationals are 
obtained from the integers ; and hence no property of numbers is 
involved beyond the fact that they form a progression. 

* See Chapter xxxvi. 



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CHAPTER XXXV. 



CANTOR'S FIRST DEFINITION OF CONTINUITY. 

271. The notion of continuity has been treated by philosophers, as 
a rule, as though it were incapable of analysis. They have said many 
things about it, including the Hegelian dictum that everything discrete 
is also continuous and vice versd^. This remark, as being an exemplifi- 
cation of Hegel's usual habit of combining opposites, has been tamely 
repeated by all his followers. But as to what they meant by continuity 
and discreteness, they preserved a discreet and continuous silence ; only 
one thing was evident, that whatever they did mean could not be 
relevant to mathematics, or to the philosophy of space and time. 

In the last chapter of Part III, we agreed provisionally to call a 
series continuous if it had a term between any two. This definition 
usually satisfied Leibniz f, and would have been generally thought 
sufficient until the revolutionary discoveries of Cantor. Nevertheless 
there was reason to surmise, before the time of Cantor, that a higher 
order of continuity is possible. For, ever since the discovery of incom- 
mensurables in Geometry — a discovery of which is the proof set forth in 
the tenth Book of Euclid — it was probable that space had continuity of 
a higher order than that of the rational numbers, which, nevertheless, 
have the kind of continuity defined in Part III. The kind which belongs 
to the rational numbers, and consists in having a term between any two, 
we have agreed to call compactness ; and to avoid confusion, I shall never 
again speak of this kind as continuity. But that other kind of con- 
tinuity, which was seen to belong to space, was treated, as Cantor 
remarks f, as a kind of religious dogma, and was exempted from that 
conceptual analysis which is requisite to its comprehension. Indeed it 
was often held to show, especially by philosophers, that any subject- 
matter possessing it was not validly analyzable into elements. Cantor 
has shown that this view is mistaken, by a precise definition of the kind 

■^ Logic f Wallace's Translation^ p. 188; Werke, v^ p. 201. 

t Phil. Werke, Gerhardt's ed.^ Vol. ii^ p. .515. But cf. Cassirer^ Leibniz System, 
Berlin, 1901, p. 183. 

\ Mannichfaltigkeitsiehre, p. 28. 



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288 Infinity and Continuity [chap, xxxy 

of continuity which must belong to space. This definition, if it is to 
be explanatory of space, must, as he rightly urges*, be effected without 
any appeal to space. We find, accordingly, in his final definition, only 
ordinal notions of a general kind, which can be fully exemplified in 
Arithmetic. The proof that the notion so defined is precisely the kind 
of continuity belonging to space, must be postponed to Part VI. Cantor 
has given his definition in two forms, of which the earlier is not purely 
ordinal, but involves also either number or quantity. In the present 
chapter, I wish to translate this earlier definition into language as 
simple and untechnical as possible, and then to show how series which 
are continuous in this sense occur in Arithmetic, and generally in the 
theory of any progression whatever. The later definition will be given 
in the following Chapter. 

272. In order that a series should be continuous, it must have two 
characteristics : it must be perfect and cohesive (zusammenhangend, 
bien enchainee)-|-. Both these terms have a technical meaning requiring 
considerable explanation. I shall begin with the latter. 

(1) Speaking popularly, a series is cohesive, or has cohesion, 
when it contains no finite gaps. The precise definition, as given by 
Cantor, is as follows : " We call T a cohesive collection of points, if for 
any two points t and t' of T, for a number e given in advance and as 
small as we please, there are always, in several ways, a finite number of 
points ^1, t^^...t^^ belonging to T, such that the distances #i, tit^^ Msv 
tj}' are all less than e.''j This condition, it will be seen, has essential 
reference to distance. It is not necessary that the collection considered 
should consist of numbers, nor that e should be a number. All that is 
necessary is, that the collection should be a series in which there are 
distances obeying the axiom of Archimedes and having no minimum, 
and that e should be an arbitrary distance of the kind presented by 
the series. If the series be the whole field of some asymmetrical 
transitive relation, or if it be the whole of the terms having a certain 
asymmetrical transitive relation to a given term, we may substitute 
stretch for distance ; and even if the series be only part of such a series, 
we may substitute the stretch in the complete series of which our series 
forms part. But we must, in order to give any meaning to cohesion, 
have something numerically measurable. How far this condition is 
necessary, and what can be done without it, I shall show at a later 
stage. It is through this condition that our discussions of quantity 
and measurement, in Part III, become relevant to the discussion of 
continuity. 



^ Acta Math, ii^ p. 403. 

t Acta Math, it, pp. 405, 406 ; Mannichfaltiglceitslehre, p. 31. 
I The words '' in several ways " seem superfluous. They are omitted by Vivanti : 
see Formulaire de Mathematiques, Vol. i, vi, § 1, No. 22. 



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271, 272] Cantor s first definition of Continuity 289 

If the distances or stretches in our series do not obey the axiom of 
Archimedes, there are some among them that are incapable of a finite 
numerical measure in terms of some others ampng them. In this case^ 
there is no longer an analogy of the requisite kind with either the 
rational or the real numbers, and the series is necessarily not 
cohesive. For let S, d be two distances ; let them be such that, for any 
finite number n^ nB is less than d. In this case, if 8 be the distance e, and 
d be the distance U\ it is plain that the condition of cohesion cannot 
be satisfied. Such cases actually occur, and — what seems paradoxical — 
they can be created by merely interpolating terms in certain cohesive 
series. For example, the series of segments of rationals is cohesive ; 
and when these segments have rational limits, the limits are not 
contained in them. Add now to the series what may be called the 
completed segments, i.e. the segments having rational limits together 
with their limits. These are new terms, forming part of the same series, 
since they have the relation of whole and part to the former terms. But 
now the difference between a segment and the corresponding completed 
segment consists of a single rational, while all other differences in the 
series consist of an infinite number of rationals. Thus the axiom of 
Archimedes fails, and the new series is not cohesive. 

The condition that distances in the series are to have no minimum is 
satisfied by real or rational numbers ; and it is necessary, if cohesion 
is to be extended to non-numerical series, that, when any unit distance 
is selected, there should be distances whose numerical measure is less 
than e, w^here e is any rational number. For, if there be a minimum 
distance, we cannot make our distances tt^^ t-J^^-" less than this minimum, 
which is contrary to the definition of cohesion. And there must not 
only be no minimum to distances in general, but there must be no 
minimum to distances from any given term. Hence every cohesive series 
must be compact, i.e. must have a term between any two. 

It must not be supposed, however, that every compact series is 
cohesive. Consider, for example, the series formed of and 2 — mjn^ 
where w, n are any integers such that rn is less than n. Here there 
is a term between any two, but the distance from cannot be 
made less than 1. Hence the series, though compact, is not co- 
hesive. This series, however, is not complete, being part only of the 
series of rationals, by means of which its distances are measured. In 
a complete series, the conditions are somewhat different. We must 
distinguish two cases, according as there are or are not distances. 
{a) If there are distances, and equal distances do not correspond to 
equal stretches, it may happen that, though the series is compact, the 
distances from some term never become less than some finite distance. 
This case would be presented by magnitudes, if we were to accept 
Meinong's opinion that the distance of any finite magnitude from zero 
is always infinite {op. cit. p. 84). It is presented by numbers, if we 
R. 19 



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290 Infinity and Continuity [chap, xxxv 

measure distances (as there are many reasons for doing) by logxly. 
Thus in this case, with regard to distances, the series is not cohesive, 
though it is complete and compact, (h) If there are no distances, 
but only stretches, then, assuming the axiom of Archimedes, any stretch 
will be less than ?ze, for a suitable value of n. Hence, dividing 
the stretch into n parts, one at least of these will be less than e. But 
there is no way of proving that all can be made less than e, unless we 
assume either the axiom of linearity (that any stretch can be divided 
into n equal parts), or a more complicated but more general axiom, to 
the effect that a stretch d can be divided into n parts, each of which is 
greater than d j {n-\-Y) and less than d j (n — l)^ whatever integer n may 
be. With this axiom and the axiom of Archimedes, a complete compact 
series must be cohesive; but these two axioms together render com- 
pleteness superfluous and compactness redundant. Thus we see that 
cohesion is in almost all cases a condition distinct from compactness. 
Compactness is purely serial, while cohesion has essential reference to 
numbers or to the conditions of numerical measurement. Cohesion 
implies compactness, but compactness never implies cohesion, except 
in the sole case of the complete series of rationals or real numbers. 

273. (S) To explain what is meant by a perfect series is more 
difficult. A series is perfect when it coincides with its first derivative*. 
To explain this definition, we must examine the notion of the derivatives 
of a series*)-, and this demands an explanation of a limiting-point of a 
series. Speaking generally, the terms of a series are of two kinds, those 
which Cantor calls isolated points, and those which he calls limiting- 
points. A finite series has only isolg^ted points ; an infinite series must 
define at least one limiting-point, though this need not belong to the 
series. A limiting-point of a series is defined by Cantor to be a term 
such that, in any interval containing the term, there are an infinite 
number of terms of the series {Ih. p. 343). The definition is given in 
terms of the points on a line, but it has no essential reference to space. 
The limiting-point may or may not be a term of the original series. 
The assemblage of all limiting-points is called the first derivative of the 
series. The first derivative of the first derivative is called the second 
derivative, and so on. Peano gives the definition of the first derivative 
of a class of real numbers as follows : Let u be a class of real numbers, 
and let ^ be a real number (which may or may not be a u) such that the 
lower limit of the absolute values of the differences of oc from terms of u 
other than x is zero ; then the class of terms oc satisfying this condition 
is the first derivative of u^. This definition is virtually identical with 
that of Cantor, but it brings out more explicitly the connection of the 
derivative with limits. A series, then, is perfect, when it consists of 

^ Acta Math, ii, p. 405. t lb. pp. 341-4. 

X Formulaire, Vol. ii. No. 3 (1899), § 71, I'O and 4-0. 



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27*2-274] Cantor s first definition of Continuity 291 

exactly the same terms as its first derivative ; i.e. when all its points are 
limiting-points, and all its limiting-points belong to it. 

274. But with regard to the latter point, namely, that all limiting- 
points of the series must belong to it, some explanation is necessary. 
Take, for example, the series of rational numbers. Every rational 
number is the limit of some series of rational numbers, and thus the 
rationals are contained in their first derivative. But as regards those 
series of rationals which do not have a rational limit, we agreed in the 
preceding chapter that they do not have a limit at all. Hence all series 
of rationals which have a limit have a rational limit, and therefore, by 
the letter of the definition, the rationals should form a perfect series. 
But this is not the case. Cantor, as we saw in connection with irrationals, 
believes, what we were compelled to regard as erroneous, that every series 
fulfilling certain conditions, which may be called the conditions of con- 
vergency, must have a limit. Hence he regards those series of rationals 
which have no rational limit as having an irrational limit, and as therefore 
having a limit not belonging to the series of rationals ; and therefore the 
series of rationals does not contain all the terms of its first derivative. 
In fact, the first derivative of the rational numbers is held to be the real 
numbers. But when we regard the real numbers as segments of rationals, 
it is impossible to take this view; and when we deny the existence- 
theorem for limits, it is necessary to modify Cantor's definition of 
perfection*. This modification we must now examine. 

What we must say is, that a series is perfect when all its points are 
limiting-points, and when further, any series being chosen out of our 
first series, if this new series is of the sort which is usually regarded as 
defining a limit, then it actually has a limit belonging to our first series. 
To make this statement precise, we must examine what are the condi- 
tions usually considered as defining a limit. In the case of denumerable 
series, they are simple, and have already been set forth. They come to 
this, that, given any distance e, however small, all the terms of our series 
after some definite term, say the mth, are such that any two of them 
have a difference whose absolute value is less than e. This statement, 
it will be seen, involves either number or quantity, i.e. it is not purely 
ordinal. It is a curious fact that, though the supposed condition for 
the existence of a limit cannot, by our present method, be stated in 
purely ordinal terms, the limit of a denumerable series, if there be one, 
can always be defined in purely ordinal terms. I shall distinguish 
Cantor's fundamental series in a compact series into progressions and 
regressions, according as earlier terms have to later ones always the 

relation P, or always the relation P (where P is the generating relation 
of the compact series in which the said progressions and regressions are 

■^ This point is ably discussed by Couturat^ Revue de Met. et de Morale, March^ 
1900, p. 167. 

19—2 



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292 Infinity and Continuity [chap, xxxy 

contained). The compact series is further assumed to be complete. A, 
term oc is then the limit of a progression, if every term of the progressio.a 
has to oc the relation P, while every term which has to x the relation P 
also has this relation to some term of the progression. This definition, 
it will be seen, is purely ordinal ; and a similar definition will apply to a 
regression. 

Let us examine next what are the usual conditions for the existence 
of a limit to a non-denumerable series. When we come to examine 
non-numerical series, we shall find it inconvenient to be restricted to 
denumerable series, and therefore it will be well to consider other series 
at once. Here, of course, if any denumerable series contained in our 
larger series fulfils the conditions for a limit, there will be a corresponding 
definition of a limiting-point in our larger series. And the upper or 
lower limit of the whole or part of our larger series, if there is one, may 
be defined exactly as in the case of a progression or a regression. But 
general conditions for the existence of a limit cannot be laid down, 
except by reference to denumerable series contained in our larger series. 
And it will be observed that Cantor's definition of a limiting-point 
assumes the existence of such a point, and cannot be turned into a 
definition of the conditions under which there are such points. This 
illustrates the great importance of Cantor's fundamental series. 

The method of segments will, however, throw some light on this 
matter. We saw in Chapter xxxiii that any class of terms in a series 
defines a segment, and that this segment sometimes can, but sometimes 
cannot, be defined by a single term. When it can be so defined, this term 
is its upper limit ; and if this term does not belong to the class by which 
the segment was defined, then it is also the upper limit of that class. 
But when the segment has no upper limit, then the class by which the 
segment was defined also has no upper limit. In all cases, however — 
and this is one of the chief virtues of segments — the segment defined by 
an infinite class which has no upper limit is the upper limit of the 
segments defined by the several members of the class. Thus, whether 
or not the class has an upper limit, the segments which its various 
terms define always have one — provided, that is, that the compact series 
in which the class is contained has terms coming after all terms oi 
the class. 

We can now express, without assuming the existence of limits in 
cases where this is not demonstrable, what is meant by a series containing 
its first derivative. When any class of terms is contained in a compact 
series, the conditions which are commonly said to insure the existence 
of an upper limit to the class, though they do not insure this, do insure 
an upper limit to the class of segments defined by the several members 
of the class. And as regards lower limits, the same proposition holds 
concerning what we called upper segments. Hence we may define : A 
class u of terms forming the whole or part of a series is perfect when 



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274, 275] Cantor s first definition of Continuity 293 

each of the terms of u is the upper or lower limit of some class contained 
in u^ and when, if v be any class contained in u^ and the lower segments 
defined by the several members of v have an upper limit, or the upper 
segments have a lower limit, this limiting segment is one of those that 
can be defined by a single term of u^ i.e. have a term of u for their 
upper or lower limit respectively. This definition, it must be admitted, 
is more complicated than Cantor's, but it is free from the unjustifiable 
assumption of the existence of limits. 

We may repeat the definition of perfection in what is perhaps less 
difficult language. Given any series, and any class of terms u contained 
in this series, there are an upper and a lower segment corresponding to 
every term of u. Any infinite set of terms v being chosen out of u^ 
there are certain conditions which are commonly said to insure that v 
has an upper limit, which, it is admitted, may belong neither to u^ nor 
to the series in which u is contained. What these conditions do insure, 
however, is that the class of lower segments corresponding to v has an 
upper limit. If the series is perfect, v will have an upper limit whenever 
the corresponding class of segments has one, and this upper limit of v 
will be a term of u. The definition of perfection requires that this 
should hold both for upper and lower limits, and for any class v con- 
tained in u. 

275. As the question concerning the existence of limits, which has 
necessitated the above complication, is one of some philosophical im- 
portance, I shall repeat the arguments against assuming the existence 
of limits in the class of series to which the rational immbers belong. 
Where a series is imperfect, while its first derivative is perfect, there 
the first derivative is logically prior to its own formation. That is to 
say, it is only by presupposing the perfect series that it can be shown 
to be the derivative of the imperfect series. We have already seen that 
this is the case with individual irrational numbers; it is easy to see 
that the principle is general. Wherever the derivative contains a term 
not belonging to the original series, that term is the limit of some 
denumerable series forming an integral part of the first series. If this 
series with a limit have the general term a^) then — wording the defi- 
nition so as not to apply only to series of numbers — there is always a 
definite number m, for any specified distance e, however small, such 
that, if n is greater than ?7z, the distance between a,n+p and an is less 
than e, whatever positive integer p may be. From this it is inferred 
that the series {a^ has a limit, and it is shown that, m many cases, 
this limit cannot belong to the series out of which the series (an) was 
chosen. But the inference that there is a limit is precarious. It may 
be supported either by previous knowledge of the term which is the 
limit, or by some axiom necessitating the existence of such a term. 
When the term which is the limit is independently known, it may be 
easily shown to be the limit. But when it is not known, it cannot be 



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294 Infinity and Contiiiuity [chap, xxxy 

proved to exist at all, unless we introduce some axiom of continuity. 
Such an axiom is introduced by Dedekind, but we saw that his axiom 
is unsatisfactory. The principle of abstraction, which shows that two 
coherent series have something in common, is fully satisfied by segments. 
And in some cases, among which is that of the rationals, it seems that 
the constitutive relation of the imperfect series cannot hold between any 
terms not belonging to this series, so that the existence of limits not 
belonging to the series is wholly impossible. For a limit must have 
a certain position in a series of which the series which it limits forms 
part, and this requires some constitutive relation of which the limit, as 
well as the terms limited, must be capable. An independent complete 
series, such as the rationals, cannot, in fact, have any limiting-points 
not belonging to it. For, if R be the constitutive relation, and two 
terms a, 5, have the relation jK, any third term c, which has this relation 
or its converse to either, and therefore both, of the terms ^, 6, belongs 
to the same series as a and h. But the limit, if it exists, must have the 
constitutive relation to the terms which it limits ; hence it must belong 
to the complete series to which they belong. Hence any series which 
has actual limiting-points not belonging to it is only part of some 
complete series ; and a complete series which is not perfect is one in 
which the limits defined in the usual way, but not belonging to the 
series, do not exist at all. Hence, in any complete series, either some 
definable limits do not exist, or the series contains its first derivative. 

In order to render the arbitrariness of assuming the existence of 
limits still more evident, let us endeavour to set up an axiom of con- 
tinuity more irreproachable than Dedekind's. We shall find that it can 
still be denied with perfect impunity. 

When a number of positions in a series continually differ less and 
less from each other, and are known to be all on one side of some given 
position, there must exist (so our axiom might run) some position to 
which they approximate indefinitely, so that no distance can be specified 
so small that they will not approach nearer than by this distance. If 
this axiom be admitted, it will follow that all imperfect series, whose 
first derivatives are perfect, presuppose these first derivatives, and are to 
be regarded as selections from them. Let us examine the consequences 
of denying our axiom in the case of a series of numbers. In this 
case, the unwary might suppose, the position next to all the terms 
cin ? but not belonging to them, would be (say) p^ where p — a^ is 
greater than e, for a suitable value of e, whatever n may be. But 
if our series is compact, there is a term between p and p — e^ say p. 
Thus p' — a^ is less than p — an, whatever n may be. Thus p^ is 
nearer all the a's than p is, contrary to the hypothesis. But the 
above denial was not direct, and the fact that it seemed correct 
illustrates the fallacies which in this subject are hard to avoid. The 
axiom is : There is a term to which the a's approach as near as we like. 



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275] Cantors first definition of Continuity 295 

The denial was : There is a term nearest to the a's, but at a finite 
distance. The denial should have been : There is no term to which 
the ^'s approach as near as we like. In other words, whatever term we 
specify, say p, there is some finite distance e, such that j9 — a^ is greater 
than e, whatever an may be. This is true in the case of series of rational 
numbers which have no rational limit. In this case, there is no term 
nearest to the a's, but at a finite distance, while also, whatever term 
beyond all the a^ we specify (except where our series has a rational 
limit), none of the cC^ approach nearer to this term than by a certain 
finite distance e. Every term beyond the a\ is at more than some 
finite distance from all of them, but there is no finite distance which 
every term beyond the a's exceeds. The introduction of irrationals 
introduces symmetry into this odd state of things, so that there is a 
term to which the a\ approach indefinitely, as well as a series of terms 
indefinitely approaching the a\s. When irrationals are not admitted, 
if we have a term p after all the a's, and a small distance e, then, if e 
be specified, p can be chosen so that p — a^ is less than e, whatever n 
may be ; but if p be specified, an e can always be found (except when 
the limit is rational) so that ^ — a^^ is greater than 6, whatever n may be. 
This state of things, though curious, is not self-contradictory. The 
admission of irrationals, as opposed to segments, is thus logically un- 
necessary ; as it is also mathematically superfluous, and fatal to the 
theory of rationals, there are no reasons in its favour, and strong reasons 
against it. Hence, finally, any axiom designed to show the existence 
of limits in cases where they cannot otherwise be shown to exist, is 
to be rejected ; and Cantor's definition of perfection must be modified 
as above. This conclusion will, in future, be regarded as established. 

Having now analyzed Cantor's earlier definition of continuity, I shall 
proceed to examine his later ordinal definition, and the application of 
its various portions to series more general than those of numbers, 
showing, if possible, the exact points in which these various portions 
are required. 



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CHAPTER XXXVI. 



ORDINAL CONTINUITY*. 



276. The definition of continuity which we examined in the pre- 
ceding chapter was, as we saw, not purely ordinal ; it demanded, in at 
least two points, some reference to either numbers, or numerically 
measurable magnitudes. Nevertheless continuity seems like a purely 
ordinal notion ; and this has led Cantor to construct a definition which 
is free from all elements extraneous to order f. I shall now examine 
this definition, as well as others which may be suggested. We shall find 
that, so long as all references to number and quantity are excluded, there 
are theorems of great importance, especially as regards fundamental 
series, which, with any suggested ordinal definition except that of 
Cantor, remain indemonstrable, and are presumably sometimes false J — 
a fact from which the merits of Cantor's definition, now to be given, are 
apparent. 

277. Cantor's definition of the continuum in his later article § is as 
follows. We start (§ 9) from the type of series presented by the rational 
numbers greater than and less than 1, in their order of magnitude. 
This type we call 77. A series of this type we define by the following 
marks. (1) It is denumerable, that is, by taking its terms in a suitable 
order (which, however, must be different from that in which they are 
given), we can give them a one-one correspondence with the finite 
integers. (S) The series has no first or last term. (3) There is a term 
between any two, Le. the series is compact {uherall dicJit). It is then 
proved that these three characteristics completely define the type of 
order presented by the rationals, that is to say, there is a one-one 
correspondence, between any two series having these three properties, in 
which earlier terms correspond to earlier terms, and later ones to later 
ones. This is established by the use of mathematical induction, which 
is applicable in virtue of the fact that series of this type are denumer- 

■^ The present chapter deals with, the same subject as M. Couturat's article^ " Sur 
la definition du Continu/' Revue de Metaphysique et de Morale, March, 1900. 1 agree 
in the main with this article^ in which much of what I said in the preceding chapter, 
and shall say in this, will be found. 

t Math, Annalen, xlvi. 

X Mathematical proofs of such theorems as are not already well known will be 
found in RdM, vii, 3. 

§ Math. Annalen, xlvi, § 11. 



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276, 277] Ordinal Continuity 297 

able. Thus all series which are denumerable, endless*, and compact, 
are ordinally similar. We now proceed (§ 10) to the consideration of 
fundamental series contained in any one-dimensional series M, We 
show (as has been already explained) what is meant by calling two 
fundamental series coherent^ and we give an ordinal definition of the 
limit of a fundamental series, namely, in the case of a progression, the 
limit comes after the whole progression, but every term before the limit 
comes before some term of the progression ; with a corresponding 
definition for the limit of a regression. We prove that no fundamental 
series can have more than one limit, and that, if a fundamental series 
has a limit, this is also the limit of all coherent series ; also that two 
fundamental series, of which one is part of the other, are coherent. Any 
term of M which is the limit of some fundamental series in M is called 
a principal term of M, If all the terms of M are principal terms, M is 
called condensed in itself (insichdicht). If every fundamental series in M 
has a limit in M^ M is called closed (abgeschlossen)f. If M is both 
closed and condensed in itself, it is perfect. All these properties, if they 
belong to M^ belong to any series which is ordinally similar to M, 
With these preparations, we advance at last to the definition of the 
continuum (§ 11). Let 6 be the type of the series to which belong the 
real numbers from to 1, both inclusive. Then ^, as we know, is a 
perfect type. But this alone does not characterize 6. It has further 
the property of containing within itself a series of the type 77, to which 
the rationals belong, in such a way that between any two terms 
of the ^-series there are terms of the ?7-series. Hence the following 
definition of the continuum : 

A one-dimensional continuum ilf is a series which (1) is perfect, 
(2) contains within itself a denumerable series aS" of which there are 
terms between any two terms of M. 

In this definition, it is not necessary to add the other properties 
which are required to show that S is of the type rj. For if S had a first 
or last term, this would be also the first or last term of M ; hence we 
could take it away from aS', and the remaining series would still satisfy 
the condition (2), but would have no first or last term ; and the 
condition (S) together with (1) insures that iS" is a compact series. 
Cantor proves that any series M satisfying the above conditions is ordi- 
nally similar to the number-continuum, i.e. the real numbers from to 1, 
both inclusive ; and hence it follows that the above definition includes 
precisely the same class of series as those that were included in his 
former definition. He does not assert that his new definition is purely 
ordinal, and it might be doubted, at first sight, whether it is so. Let 
us see for ourselves whether any extra-ordinal notions are contained in it. 

■^ I.e. having- neither a beginning- nor an end. 

t Not to be confounded with the elementary sense of a closed series discussed in 
Part IV. 



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298 Infiiiity and Continuity [chap, xxxvi 

278. The only point as to which any doubt could arise is with 
regard to the condition of being denumerable. To be a denumerable 
collection is to be a collection whose terms are all the terms of some 
progression. This notion, so far, is purely ordinal. But in the case 
supposed, that of the rationals or of any ordinally similar series, the 
terms forming the series must be capable of two orders, in one of which 
they form a compact series, while in the other they form a progression. 
To discover whether or not a given set of terms is capable of these two 
orders, will in general demand other than ordinal conditions ; never- 
theless, the notion itself is purely ordinal. Now we know, from the 
similarity of all such series to the series of rationals (which involves only 
ordinal ideas), that no such series is perfect. But it remains to be seen 
whether we can prove this without appealing to the special properties 
of the rationals which result from there being a series in which there is 
distance. We know, as a matter of fact, that no denumerable series can 
be perfect*, but we want here a purely ordinal proof of this theorem. 
Such a proof, however, is easily given. For take the terms of our 
denumerable compact series S in the order in which they form a 
progression, and in this order call them u. Starting with the first in 
this order, which we will call x^^ there must be one which, in the other 
order S^ follows this term. Take the first such term, x\^ as the second 
in a fundamental series v. This term has a finite number of predecessors 
in the progression u^ and therefore has successors in S which are also 
successors in u ; for the number of successors in S is always infinite. 
Take the first of these common successors, say ^25 ^s the third term of 
our fundamental series v. Proceeding in this way, we can construct an 
ascending fundamental series in S^ the terms of which have the same 
order in u as in S. This series cannot have a limit in S^ for each term x^ 
succeeds, in S^ every term which precedes it in u. Hence any term of S-^ 
will be surpassed by some term x^ of our fundamental series v^ and 
hence this fundamental series has no limit in S. The theorem that a 
denumerable endless series cannot be perfect is, therefore, purely ordinal. 
From this point onwards there is no difficulty, and our former theory of 
segments enables us to state the matter simply. Given a denumerable, 
endless, compact series S^ construct all the segments defined by funda- 
mental series in S, These form a perfect series, and between any two 
terms of the series of segments there is a segment whose upper (or 
lower) limit is a term of S. Segments of this kind, which may be called 
rational segments, are a series of the same type as S^ and are contained 
in the whole series of segments in the required manner. Hence the 
ordinal definition of the continuum is complete. 

279. It must not be supposed that continuity as above defined can 
only be exemplified, in Arithmetic, by the devious course from integers 
to rationals, and thence to real numbers. On the contrary, the integers 

■^ Acta Mathematica, 11^ p. 409. 



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278-280] Ordinal Continuity 299 

themselves can be made to illustrate continuity. Consider all possible 
infinite classes of integers, and let them be arranged on the following 
plan. Of two classes u^ v^ of which the smallest number in ic is less than 
the smallest in v, u comes first. If the first n terms of u and v are 
identical, but the (n + iy^ terms are different, that which has the 
smaller (n + lf^ term is to come first. This series has a first term, 
namely, the whole class of the integers, but no last term. Any completed 
segment of the series, however, is a continuous series, as the reader can 
easily see for himself. The denumerable compact series contained in it 
is composed of those infinite classes which contain all numbers greater 
than some number, i.e, those containing all but a finite number of 
numbers. Thus classes of finite integers alone suffice to generate con- 
tinuous series. 

280. The above definition, it will be observed, depends upon pro- 
gressions. As progressions are the very essence of discreteness, it seems 
paradoxical that we should require them in defining continuity*. And, 
after all, as it is certain that people have not in the past associated any 
precise idea with the word continuity^ the definition we adopt is, in some 
degree, arbitrary. Series having the properties enumerated in Cantor's 
definition would generally be called continuous, but so would many others 
which his definition excludes. In any case it will be a valuable inquiry 
to ask what can be done by compact series without progressions. 

Let u be any endless compact series, whose generating relation is P, 
and concerning which nothing further is known. Then, by means of any 
term or any class of terms in u^ we can define aj segment of u. Let us 
denote by U the class of all lower segments of u. A low^r segment, it 
may be well to repeat, is a class v of terms contained in u^ not null, 
and not coextensive with ?^, and such that v has no last term, and 
every term preceding a t^ is a , f . In the converse case, when v has 
no first term, and every term following a v is a v^ v is called an 
upper segment. It is then easy to prove that every segment consists 
of all the terms preceding (or following) either some single term of u^ or 
a variable term of some class of terms oi u-^ and that every single term, 
and every class of terms, defines an upper and a lower segment in this 
manner. Then, if V denote the class of upper segments, it is easy to 
prove that both U and V are again endless compact series, whose 
generating relation is that of whole or part ; while if u has one or two 
ends, so have U and F, though the end-terms are not segments according 
to the definition. If we now proceed to the consideration of segments 

* Mr Whitehead has shown that the following simpler definition is equivalent 
to Cantor's. A series is continuous when (1) every segment^ upper or lower^ has a 
limits and the series has a first and a last term ; (2) a d enumerable compact series is 
contained in it in such a way that there are ^erms of this latter series between any 
two terms of our original series. In this definition, progressions are relevant only 
in defining a denumerable series. 



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300 Infinity and Continuity [chap, xxxvi 

in U or F (C7 say), we shall find that the segment of W^ defined by any 
class whatever of W^ can always be defined by a single f/, which, if the 
class is infinite and has no last term, is the upper limit of the class, and 
which, in all cases, is the logical sum of all the members of the class — 
members which, it must be remembered, are all themselves classes 
contained in u^. Hence all classes contained in U and having no last 
term have an upper limit in U ; and also (what is a distinct proposition) 
all classes contained in U and having no first term have a lower limit 
in f/, except in the case where the lower limit is the logical zero or 
null-class ; and the lower limit is always the logical product of all the 
classes composing the class which it limits. Thus by adding to U the 
null-class, we insure that U shall be a closed series. There is a sense 
in which U is condensed in itself, namely, this : every term of U is the 
upper limit of a suitably chosen class contained in C, for every term is 
the upper limit of the segment of ^'s which it defines ; and every term 
of 77 is a lower limit of the class of those f/'s of which it is a proper 
part. But there is absolutely no proof, so far at least as I have been 
able to discover, that every term of U is the upper or lower limit of a 
fundamental series. There is no a priori reason why, in any series, the 
limit of any class should always be also the limit of a fundamental 
series ; this seems, in fact, to be a prerogative of series of the types to 
which rationals and real numbers respectively belong. In our present 
case, at least, though our series is, in the above general sense, condensed 
in itself, there seems no reason for supposing its terms to be all of them 
limits of fundamental series, and in this special sense the series may not 
be condensed in itself. 

281. It is instructive to examine the result of confining the terms 
of U to such segments as can be defined by fundamental series. In this 
case it is well to consider, in addition to upper and lower segments, their 
supplements, as they may be called, of which I shall shortly give the 
definition. Let a compact series v be given, generated by a transitive 
asymmetrical relation P, and let u be any fundamental series in v. If 
earlier terms of u have to later ones the relation P, I shall call u a 
progression ; if the relation P, I shall call u a regression. If now i<d be 
any class whatever contained in v^ w defines, as we have already seen, 
four other classes in ty^namely (1) the class of terms before every w, 
which I shall call ivtt ; (2) the class of terms after every %€., which I shall 
call WTT ; (3) the class of terms before some ic, which I shall call ttio ; 
(4) the class of terms after some re^, which I shall call rirw. The classes 
(3) and (4) are lower and upper segments respectively ; the classes (1) and 

■^ The definition of the logical sum of the members of a class of classes^ in a form 
not involving iinitude^ is^ I believe^ due to Peano. It is as follows : Let w be a class 
of classes ; then the logical sum of the members of w is the class of terms oc such that 
there is some class belonging to w, to which oc belongs. See Formulaire, Vol. ii, 
Part I (1897), No. 461. 



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280, 281] Ordinal Continuity 301 

(2) are supplements to (4) and (3) respectively, and I shall call them 
supplemental segments. When lo has an upper limit, this is the first term 
of WTT^ and thus wtt is not a segment, since no upper segment has a first 
term. But when xv has no upper limit, then, whether w be finite or 
infinite, wtt is a segment. Similar remarks apply to lower limits. If w 
has a last term, this belongs neither to irto nor to wir^ but all other terms 
of V belong to one or other class ; if to has no last term, all terms of v 
belong to ttw or zott. Similar remarks apply to wtt and iriv. Applying 
these general definitions to the cases of progressions and regressions, 
we shall find that, for a progression, only the classes (2) and (3) are 
important ; for a regression, only the classes (1) and (4). The question 
where a progression begins or a regression ends is quite unimportant. 
Since a progression has no last term, and a regression no first term, 
the segment defined by either, together with its supplement, contains 
every term of v. Whether progressions and regressions in v have limits 
always, sometimes, or never, there seems no way of deciding from the 
given premisses. I have not been able to discover an instance of a 
compact series where they never have limits, but I cannot find any proof 
that such an instance is impossible. 

Proceeding now to classes of segments, as we proceeded before to our 
class f/, we have here four such classes to consider, namely : (1) The class 
viT^ each of whose terms is the class uir defined by some regression Uy 
i.e.^ the terms of v which come before all the terms of some regression in 
V ; {^) the class i'tt, consisting of all the classes utt defined by pro- 
gressions u ; (3) the class 7rf , whose terms are irii^ where ii is some 
progression ; (4) the class vir^ whose terms are uir^ where u is some 
regression. Each of these four classes is a class of classes, for its terms 
are classes contained in v. Each of the four is itself a compact series. 
There is no way of proving, so far as I know, that (1) and (3), or (2) 
and (4), have any common terms. Each pair would have a common 
term if v contained a progression and a regression which were coherent, 
and had no limit in v^ but there is no way of discovering whether this 
case ever arises in the given series v. 

When we come to examine whether the four classes thus defined are 
condensed in themselves, we obtain the most curious results. Every 
fundamental series in any one of the four classes has a limit, but not 
necessarily in the series of which its terms are composed, and conversely, 
every term of each of our four classes is the limit of a fundamental 
series, but not necessarily of a series contained in the same class to which 
the limiting term belongs. The state of things, in fact, is as follows : 

Every progression in vtt or irv has a limit in irv. 

Every progression in vtt or irv has a limit in ttv. 

Every regression in vir or ttv has a limit in vtt. 

Every regression in vtt or ttv has a limit in vtt. 

Every term of vtt is the limit of a regression in vtt and of one in ttv. 



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302 Infinity and Continuity [chap, xxxyi 

Every term of vir is the limit of a regression in vir and of one in irv. 

Every term of ttv is the limit of a progression in vtt and of one 
in iTV, 

Every term of irv is the limit of a progression in vir and of one 
in TTV. 

Hence vir is identical with the class of limits of regressions in vir or irv ; 

VTT is identical with the class of limits of regressions in vnr or irv ; 

irv is identical with the class of limits of progressions in vtt or irv ; 

TTV is identical with the class of limits of progressions in ttv or vnr. 

Thus each of our four classes has a kind of one-sided perfection ; 
two of the four are perfect on one side, the other two on the other. 
But I cannot prove of any one of the four classes that it is wholly 
perfect. We might attempt the combination of vtt and ttv^ and also of 
VTT and ttv. For vtt and ttv together form one series, whose generating 
relation is still whole and part. This series will be perfect, and will 
contain the limits alike of progressions and of regressions in itself. But 
this series may not be compact ; for if there be any progression a and 
regression ii in ^, which both have the same limit in v (a case which, as 
we know, occurs in some compact series), then ttu and iltt will be 
consecutive terms of the series formed of ttv and vtt together, for h'tt 
will contain the common limit, while ttu will not contain it, but 
all other terms of v will belong to both or to neither. Hence when our 
series is compact, we cannot show that it is perfect ; and when we have 
made it perfect, we can show that it may not be compact. And a 
series which is not compact can hardly be called continuous. 

Although we can prove that, in our original compact series v^ there 
are an infinite number of progressions coherent with a given progression 
and having no term in common with it, we cannot prove that there is 
even one regression coherent with a given progression ; nor can we 
prove that any progression or regression in v has a limit, or that 
any term of t; is a limit of a progression or regression. We cannot 
prove that any progression u and regression u are such that ttu = iltt^ 
nor yet that ttu and iltt may diflPer by only a single term of v. Nor, 
finally, can we prove that any single progression in vtt has a limit in vtt^ 
with similar propositions concerning the other three classes i^tt, ttv^ ttv. 
At least, I am unable to discover any way of proving any of these 
theorems, though in the absence of instances of the falsity of some 
of them it seems not improbable that these may be demonstrable. 

If it is the fact — as it seems to be — that, starting only from a 
compact series, so many of the usual theorems are indemonstrable, 
we see how fundamental is the dependence of Cantor's ordinal theory 
upon the condition that the compact series from which we start is to be 
denumerable. As soon as this assumption is made, it becomes easy 
to prove all those of the above propositions which hold concerning the 
types 7] and 6 respectively. This is a fact which is obviously of con- 



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281, 282] Ordinal Continuity 303 

siderable philosophical importance ; and it is with a view of bringing it 
out clearly that I have dwelt so long upon compact series which 
are not assumed to be denumerable. 

282. The remark which we made just now, that two compact 
series may be combined to form one which sometimes has consecutive 
terms, is rather curious, and applies equally to continuity as defined by 
Cantor. Segments of rationals form a continuous series, and so do 
completed segments {ix. segments together with their limits) ; but the 
two together form a series which is not compact, and therefore not 
continuous. It is certainly contrary to the usual idea of continuity 
that a continuous series should cease to be so merely by the interpolation 
of new terms between the old ones. This should, according to the usual 
notions, make our series still more continuous. It might be suggested 
that, philosophically speaking, a series cannot be called continuous unless 
it is complete^ i.e. contains a certain term together with all the terms 
having to the given term a specified asymmetrical transitive relation or its 
converse. If we add this condition, the series of segments of rationals is 
not complete with regard to the relation by which we have hitherto 
regarded it as generated, since it does not consist of all classes of 
rationals to which a given segment has the relation of whole and part, 
and each of which contains all terms less than any one of its terms — 
this condition is also satisfied by completed segments. But every series 
is complete with regard to some relation, simple or complex. This is 
the reason why completeness need not, from a mathematical standpoint, 
be mentioned in the definition of continuity, since it can always be 
insured by a suitable choice of the generating relation. 

We have now seen in what Cantor's definition of continuity consists, 
and we have seen that, while instances fulfilling the definition may 
be found in Arithmetic, the definition itself is purely ordinal — the only 
datum required is a denumerable compact series. Whether or not the 
kind of series which Cantor defines as continuous is thought to be the 
most similar to what has hitherto been vaguely denoted by the word, 
the definition itself, and the steps leading to it, must be acknowledged 
to be a triumph of analysis and generalization. 

Before entering upon the philosophical questions raised by the 
continuum, it will be well to continue our review of Cantor's most 
remarkable theorems, by examining next his transfinite cardinal and 
ordinal numbers. Of the two problems with which this Part is 
concerned, we have hitherto considered only continuity ; it is now time 
to consider what mathematics has to say concerning infinity. Only 
when this has been accomplished, shall we be in a position adequately 
to discuss the closely allied philosophical problems of infinity and 
continuity. 



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CHAPTER XXXVII. 



THANSFINITE CARDINALS. 

283. The mathematical theory of infinity may ahiiost be said 
to begin with Cantor. The Infinitesimal Calculus, though it cannot 
wholly dispense with infinity, has as few dealings with it as possible, 
and contrives to hide it away before facing the world. Cantor has 
abandoned this cowardly policy, and has brought the skeleton out of its 
cupboard. He has been emboldened in this course by denying that it 
is a skeleton. Indeed, like many other skeletons, it Avas wholly 
dependent on its cupboard, and vanished in the light of day. Speaking 
without metaphor. Cantor has established a new branch of Mathematics, 
in which, by mere correctness of deduction, it is shown that the 
supposed contradictions of infinity all depend upon extending, to the 
infinite, results which, while they can be proved concerning finite 
numbers, are in no sense necessarily true of all numbers. In this theory, 
it is necessary to treat separately of cardinals and ordinals, which are 
far more diverse in their properties when they are transfinite than when 
they are finite. Following the same order as previously — the order 
which seems to me to be alone philosophically correct — I shall begin 
with transfinite cardinals*. 

284. The transfinite cardinals, which are also called powers^ may be 
defined in the first place so as to include the finite cardinals, leaving it 
to be investigated in what respects the finite and the transfinite are 
distinguished. Thus Cantor gives the following definition^. 

" We call the power or cardinal number of M that general idea 
which, by means of our active faculty of thought, is deduced from the 
collection M^ by abstracting from the nature of its diverse elements and 
from the order in which they are given." 

This, it will be seen, is merely a phrase indicating what is to be 
spoken of, not a true definition. It presupposes that every collection 

■^ This is the order followed in Math, Annalen, xlvi_, but not in the Mannich- 
faltigkeitslehre. 

t Math. Annalen, xlvi^ § 1. 



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283, 284] Transfinite Cardinals 305 

has some sucli property as that indicated — a property, that is to 
say, independent of the nature of its terms and of their order ; depend- 
ing, we might feel tempted to add, only upon their number. In fact, 
number is taken by Cantor to be a primitive idea, and it is, in his theory, 
a primitive proposition that every collection has a number. He is 
therefore consistent in giving a specification of number which is not a 
formal definition. 

By means, however, of the principle of abstraction, we can give, as 
we saw in Part II, a formal definition of cardinal numbers. This 
method, in essentials, is given by Cantor immediately after the above 
informal definition. We have already seen that, if two classes be called 
similar when there is a one-one relation which couples every term of 
either with one and only one term of the other, then similarity is sym- 
metrical and transitive, and is reflexive for all classes. A one-one relation, 
it should be observed, can be defined without any reference to number, 
as follows : A relation is one-one when, if oc has the relation to «/, and x 
differs from x^ y' from ?/, then it follows that x' does not have the 
relation to ?/, nor x to y . In this there is no reference to number ; and 
the definition of similarity also is therefore free from such reference. 
Since similarity is reflexive, transitive, and symmetrical, it can be 
analyzed into the product of a many-one relation and its converse, and 
indicates at least one common property of similar classes. This property, 
or, if there be several, a certain one of these properties, we may call the 
cardinal number of similar classes, and the many-one relation is that of 
a class to the number of its terms. In order to fix upon one definite 
entity as the cardinal number of a given class, we decide to identify the 
number of a class with the whole class of classes similar to the given 
class. This class, taken as a single entity, has, as the proof of the 
principle of abstraction shows, all the properties required of a cardinal 
number. The method, however, is philosophically subject to the doubt 
resultiii^^ from the contradiction set forth in Part I, Chapter x.* 

In this way we obtain a definition of the cardinal number of a class. 
Since similarity is reflexive for classes, every class has a cardinal number. 
It migttt be thought that this definition would only apply to finite 
classes, since, to prove that all terms of one class are correlated with all 
of another, complete enumeration might be thought necessary. This, 
however, is not the case, as may be seen at once by substituting any for 
all — a word which is generally preferable where infinite classes are 
concerned. Two classes u^ v are similar when there is some one-one 
relation II ; iich that, if x be any z/, there is some term y o( v such that 
xRy \ and it y' be any v^ there is some term x of u such that xRy. 
Here there is no need whatever of complete enumeration, but only of 
propositions concerning any u and any v. For example, the points on 
a p^iven line are similar to the lines through a given point and meeting 

^ See Appendix. 
E. 20 



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306 Infinity and Continuity [chap, xxxvii 

the given line ; for any point on the given line determines one and only 
one line through the given point, and any line through the given point 
meeting the given line determines one and only one point on the given 
line. Thus where our classes are infinite, we need some general 
proposition about any term of either class to establish similarity, but 
we do not need enumeration. And in order to-prove that every (or any) 
class has a cardinal number, we need only the observation that any term 
of any class is identical with itself. No other general proposition about 
the terms of a class is requisite for the reflexive property of similarity. 

285. Let us now examine the chief properties of cardinal numbers. 
I shall not give proofs of any of these properties, since I should merely 
repeat what has been said by Cantor. Considering first their relations 
to classes, we may observe that, if there be two sets of classes which are 
similar in pairs, and no two of the one set have any common part, nor 
yet any two of the other set, then the logical sum of all the classes of 
one set is similar to the logical sum of all the classes of the other set. 
This proposition, familiar in the case of finite classes, holds also of 
infinite classes. Again, the cardinal number of a class u is said to be 
greater than that of a class v^ when no part of v is similar to u^ but 
there is ^^Ipart of u which is similar to v. In this case, also, the number 
of V is r, lid to be less than that of ii. It can be proved that, if there 
is a pa.Dt of u which is similar to v^ and a part of v which is similar 
to u^ then u and v are similar*. Thus equal, greater, and less are all 
incompatible with each other, all transitive, and the last two asym- 
metrical. We cannot prove at all simply — and it seems more or less 
doubtful whether we can prove at all — that of two different cardinal 
numbers one must be greater and the other lessf. It is to be observed 
that the definition of greater contains a condition not required in the 
case of finite cardinals. If the number of v be finite, it is sufficient 
that a proper part of u should be similar to v. But among transfinite 
cardinals this is not sufficient. For the general definition of greater^ 
therefore, both parts are necessary. This difference between finite 
and transfinite cardinals results from the defining difference of finite 
and infinite, namely that when the number of a class is not finite, 
it always has a proper part which is similar to the whole ; ( that is, 
every infinite class contains a part (and therefore an infinite/ number 
of parts) having the same number as itself. Certain particulai- cases of 
this proposition have long been known, and have been regarded as con- 
stituting a contradiction in the notion of infinite number. ILeibniz, for 
example, points out]: that, since every number can be doubled, the 

■^ Bernstein and Schroder's theorem ; for proofs see Borel^ Let^onsdzir la theorie 
des fonctions, Paris^ 1898^ Note I^ and Zermelo^ Gottinger Nachrichten, 1901^ 
pp. 34—38. 

+ Cantor's grounds for holding that this is so are vague^ and do not appear, to 
me to be valid. They depend upon the postulate that every class is the field of so^ 
well-ordered relation. See Cantor^ Math. Annalen, xini, note to § 2. 

X Gerhardt's ed. I, p. 338. 



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284-286] Transfinite Cardinals 30T 

number of numbers is the same as the number of even numbers, whence 
he deduces that there is no such thing as infinite number. The first to 
generahze this property of infinite collections, and to treat it as not 
contradictory, was, so far as I know, Bolzano*. But the strict proof 
of the proposition, when the finite cardinals are defined by means of 
mathematical induction, as well as the demonstration that it is not 
contradictory, are due to Cantor and Dedekind. The proposition itself 
may be taken as the definition of the transfinite among cardinal numbers, 
for it is a property belonging to all of them, and to none of the finite 
cardinals f. Before examining this property further, however, we must 
acquire a more intimate acquaintance with the other properties of 
cardinal numbers. 

286. I come now to the strictly arithmetical properties of cardinals, 
i.e. their addition, multiplication, etc.:|:. The addition of numbers is 
defined, when they are transfinite, exactly as it was defined in the case 
of finite numbers, namely by means of logical addition. The number 
of the logical sum of two classes which have no common term is the 
sum of the numbers of the two classes. This can be extended by suc- 
cessive steps to any finite number of classes ; for an infinite number of 
classes, forming a class of classes, the sum of their number no two 
have any common term, is still the number of their logical . a — and 
the logical sum of any class of classes, finite or infinite, is logically 
definable. For sums of two or three numbers, so defined, the com- 
mutative and associative laws still hold, i.e. we still have 

a-}-b=^b + a and a + (b -{- c) = (a -^ b) + c. 
The multiplication of two numbers is thus defined by Cantor : If M 
and N be two classes, we can combine any element of M with any 
element of N to form a couple (t/z, n) ; the number of all such couples 
is the product of the numbers of M and iV. If we wish to avoid the 
notion of a couple in the definition, we may substitute the following § : 
Let u be a class of classes, a in number ; let each of these classes be- 
longing to u contain b terms ; and let no two of these classes have an; 
common term ; then ab is the number of the logical sum of all these 
classes. This definition is still purely logical, and avoids the notion of 
a couple. Multiplication so defined obeys the commutative, associative, 
and distributive laws, i.e. we have 

ab = ba, a (be) = (ab) c, a(b + c) = ab-h ac. 
Hence addition and multiplication of cardinals, even when these are 
transfinite, satisfy all the elementary rules of Arithmetic. 

■^ Paradoooien des Unendliche7iy § 21. 

t See Dedekind^ Was sind und was sollen die Zahlen ? No. 64. 

J Cantor^ Math. Annalen, xlvi^ § 3 ; Whitehead^ American Journal of Math. 
Vol. XXI v^ No. 4. 

§ Vivanti^ Theoiie des Ensembles, Formulaire de Mathematiques, Vol. i, Part yi, 
§ 2, No. 4. 

20—2 



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308 Infinity and Continuity [chap, xxxyii 

The definition of powers of a number {o?) is also effected logically 
{ih. § 4). For this purpose. Cantor first defines what he calls a covering 
{Belegung) of one class N by another M. This is a law by which, to 
every element n of iVis joined one and only one element m of M^ but 
the same element m may be joined to many elements of iV^ That is, 
a Belegung is a many-one relation, whose domain includes lY, and 
which correlates with the terms of N always terms of M. If a be the 
number of terms in M^ b the number in iV, then the number of all such 
many-one relations is defined to be aP. It is easy to see that, for finite 
numbers, this definition agrees with the usual one. For transfinite 
numbers, indices have still the usual properties, i.e. 
ot^a^ = a^ + % a«&« = (a6) % {of") ^-=aP\ 

In the case where « = 2, aP is capable of a simpler definition, deduced 
from the above. If a = 2, & will be the number of ways in which 
h terms can be related each to one of two terms. Now when those 
which are related to one of the two are given, the rest are related to 
the other. Hence it is enough, in each case, to specify the class of 
terms related to one of the two. Hence we get in each case a class 
chosen out of the h terms, and in all cases we get all such classes. 
Hence & is the number of classes that can be formed out of h terms, 
or the number of combinations of h things any number at a time — a 
familiar theorem when h is finite, but still true when h is transfinite. 
Cantor has a proof that 2^ is always greater than h — a proof which, 
however, leads to difficulties when h is the number of all classes, or. 
more generally, when there is some collection of h terms in which all the 
sets chosen out of the h terms are themselves single terms of &*. 

The definitions of multiplication given by Cantor and Vivanti re- 
quire that the number of factors in a product should be finite ; and 
this makes it necessary to give a new and independent definition of 
powers, if the exponent is allowed to be infinite. Mr A. N. Whitehead "f 
has given a definition of multiplication which is free from this restriction, 
and therefore allows powers to be defined in the ordinary way as pro- 
ducts. He has also found proofs of the formal laws when the number 
of summands, brackets, or factors is infinite. The definition of a product 
is as follows : Let A; be a class of classes, no two of which have any terms 
in common. Choose out, in every possible way, one term and only one 
from each of the classes composing Jc. By doing this in all possible 
ways, we get a class of classes, called the multiplicative class of h. The 
number of terms in this class is defined to be the product of the numben 
of terms in the various classes which are members of h. Where h haj 
a finite number of members, it is easy to see that this agrees with the 
usual definition. Let u^ v^ w be the members of A;, and let them have 
respectively a, /3, 7 terms. Then one term can be chosen out of u ir 

■^ See Chapter xliii_, infra. t American Journal of Mathematics^ loc. cit. 



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286, 287] Transjinite Cardinals 309 

a ways : for every way there are y3 ways of choosing one term out of v , 
and for every way of choosing one term out of u and one out of v, there 
are 7 ways of choosing one out of i<o. Hence there are a ^ 7 ways of 
choosing one term out of each, when multiphcation is understood in its 
usual sense. The multiplicative class is an important notion, by means of 
which transfinite cardinal Arithmetic can be carried a good deal further 
than Cantor has carried it. 

287. All the above definitions apply to finite and transfinite integers 
alike, and, as we see, the formal laws of Arithmetic still hold. Trans- 
finite integers differ from finite ones, however, both in the properties of 
their relation to the classes of which they are the numbers, and also in 
regard. to the properties of classes of the integers themselves. Classes of 
numbers have, in fact, very different properties according as the numbers 
are all finite or are in part at least transfinite. 

Among transfinite cardinals, some are particularly important, 
especially the number of finite numbers, and the number of the con- 
tinuum. The number of finite numbers, it is plain, is not itself a 
finite number ; for the class Jinite number is similar to the class even 
finite number^ which is a part of itself. Or again the same conclusion 
may be proved by mathematical induction — a principle which also 
serves to define finite numbers, but which, being of a more ordinal 
nature, I shall not consider until the next chapter. The number of 
finite numbers, then, is transfinite. This number Cantor denotes by 
the Hebrew Aleph with the suffix ; for us it will be more convenient 
to denote it by a^. Cantor proves that this is the least of all the 
transfinite cardinals. This results from the following theorems (loo. 
cit. § 6) : 

(A) Every transfinite collection contains others as parts whose 
number is (Xq. 

(B) Every transfinite collection which is part of one whose 
number is ao, also has the number a^. 

(C) No finite collection is similar to any proper part of itself. 

(D) Every transfinite collection is similar to some proper part of 
itself*. 

From these theorems it follows that no transfinite number is less than 
the number of finite numbers. Collections which have this number are 
said to be denumerable, because it is always possible to cou7it such 
collections, in the sense that, given any term of such a collection, there 
is some finite number n such that the given term is the r^th. This 
is merely another way of saying that all the terms of a denumerable 
collection have a one-one correlation with the finite numbers, which again 
is equivalent to saying that the number of the collection is the same 
as that of the finite numbers. It is easy to see that the even numbers, 
the primes, the perfect squares, or any other class of finite numbers 

•^ Theorems G and D require that the finite should be defined by mathematical 
induction^ or else they become tautolog-ous. 



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310 Infinity and Continuity [chap, xxxvii 

having no maximum, will form a denumerable series. For, arranging 
any such class in order of magnitude, there will be a finite number of 
terms, say n^ before any given term, which will thus be the {n-\-\)th 
term. What is more remarkable is, that all the rationals, and even 
all real roots of equations of a finite degree and with rational co- 
efficients {i.e. all algebraic numbers), form a denumerable series. And 
even an 72-dimensional series of such terms, where n is a finite number, 
or the smallest transfinite ordinal, is still denumerable*. That the 
rational numbers are denumerable can be easily seen, by arranging them 
in the order in which those with smaller sum of numerator and denomi- 
nator precede those with larger sum, and of those with equal sums, those 
with the smallerniipierators precede those with larger ones. Thus we 
get the series 

1, 1/2, 2, 1/3, 3, 1/4, 2/3, 3/2, 4, 1/5 ... 

This is a discrete series, with a beginning and no end ; every rational 
number will occur in this series, and will have a finite number of pre- 
decessors. In the other cases the proof is rather more difficult. 

All denumerable series have the same cardinal number ao, however 
different they may appear. But it must not be supposed that there is 
no number greater than a^. On the contrary, there is an infinite series 
of such numbers f. The transfinite cardinals are asserted by Cantor to 
be well-ordered, that is, such that every one of them except the last of 
all (if there be a last) has an immediate successor, and so has every 
class of them which has any numbers at all after it. But they do not 
all have an immediate predecessor ; for example, a^ itself has no imme- 
diate predecessor. For if it had one, this would have to be the last of 
the finite numbers; but we know that there is no last finite number. But 
Cantor's grounds for his assertion that the cardinals are well-ordered seem 
insufficient, so that for the present this must remain an open question. 

288. Of the transfinite numbers other than ao, the most important 
is the number of the continuum. Cantor has proved that this number 
is not ao|j ^i^d hopes to prove that it is ai§ — a hope which, though 
he has long cherished it, remains unfulfilled. He has shown that the 
number of the continuum is ^f^'^W — a most curious theorem ; but it 
must still remain doubtful whether this number is aj, though there 
are reasons which rendered this probable If. As to the definition of a^ 

■^ See Acta Mathematica, ii, pp. 306^ 813^, 326. 

t See Jahreshericht der deutscheii Mathematiker-Vereinigung 1, 1892; Rivista di 
Matematica, 11^ pp. 165-7. Cantor's assertion that there is no greatest transfinite 
cardinal is open to question. See Chap. xliii_5 infra. 

I Acta Math. 11^ p. 308. § lb. p. 404. a^ is the number next after Qq. 

II Math. Annalen^ xlvi, § 4^ note. 

IT See Couturat^ De I'Injini Mathematique, Paris^ 1896, p. ^66. The ground 
alleged by Cantor for identifying the second power with that of the continuum is^ 
that every infinite linear collection of points has either the first power, or that of the 
continuum, whence it would seem to follow that the power of the continuum must 
be the next after the first. {Math. Annalen, 23, p. 488 ; see also Acta Math, vii.) But 



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287-289] Transfinite Cardinals 311 

and of the whole succession of transfinite cardinals, this is a matter 
which is better postponed until we have discussed the transfinite ordinals. 
It must not be supposed that we can obtain a new transfinite cardinal 
by merely adding one to it, or even by adding any finite number or a^. 
On the contrary, such puny weapons cannot disturb the transfinite 
cardinals. It is known that in the case of ao and a certain class of trans- 
finite cardinals, a number is equal to its double ; also that in the case of 
tto and a presumably different class of transfinite cardinals, a number is 
equal to its square. The sum of two numbers belonging to the former 
of these classes is equal to the greater of the two numbers. It is not 
known whether all transfinite cardinals belong to one or both of these 
classes *. 

289. It may be asked : In what respect do the finite and transfinite 
cardinals together form a single series ? Is not the series of finite 
numbers complete in itself, without the possibility of extending its 
generating relation ? If we define the series of integers by means of 
the generating relation of differing by one — the method which is 
most natural when the series is to be considered as a progression — 
then, it must be confessed, the finite integers form a complete series, 
and there is no possibility of adding terms to them. But if, as is 
appropriate in the theory of cardinals, we consider the series as arising 
by correlation with that of whole and part among classes of which the 
integers can be asserted, then we see that this relation does extend 
beyond finite numbers. There are an infinite number of infinite classes 
in which any given finite class is contained; and thus, by correlation 
with these, the number of the given finite class precedes that of any one 
of the infinite classes. Whether there is any other sense in which all 
integers, finite and transfinite, form a single series, I leave undecided ; 
the above sense would be sufficient to show that there is no logical 
error in regarding them as a single series, if it were known that of 
any two cardinals one must be the greater. But it is now time to turn 
our attention to the transfinite ordinals. 

the inference seems somewhat precarious. Consider^ for example^ the following 
analogy : in a compact series^ the stretch determined by two terms consists either of 
an infinite number of terms^ or_, when the two terms coincide^ of one term only^ and 
never of a finite number of terms other than one. But finite stretches are presented 
by other types of series^ e.g. progressions. 

The theorem that the number of the continuum is 2«o results very simply from the 
proposition of Chapter xxxvi^ that infinite classes of finite integers form a continuous 
series. The number of all classes of finite integers is 2^0 {vide supra), and the number of 
finite classes is aq . Hence the number of all infinite classes of finite integers is 2«o for 
the subtraction of qq does not diminish any number greater than a^ ; 2*^0 is therefore 
the number of the continuum. To prove that this number is a^ it would therefore 
be sufficient to show that the number of infinite classes of finite integers is the same 
as the number of types of series that can be formed of all the finite integers ; for the 
latter number, as we shall see in the next chapter, is a^. 

* Cf. Whitehead, /oc. cif. pp. 392-4. 



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CHAPTER XXXVIII. 



TRANSFINITE ORDINALS. 

290. The transfinite ordinals are, if possible, even more interesting 
and remarkable than the transfinite cardinals. Unlike the latter, they 
do not obey the commutative law, and their arithmetic is therefore quite 
different from elementary arithmetic. For every transfinite cardinal, or 
at any rate for any one of a certain class, there is an infinite collection 
of transfinite ordinals, although the cardinal number of all ordinals is 
the same as or less than that of all cardinals. The ordinals which 
belong to series whose cardinal number is a^ are called the second class 
of ordinals ; those corresponding to a^ are called the third class, and 
so on. The ordinal numbers are essentially classes of series, or better 
still, classes of generating relations of series ; they are defined, for the 
most part, by some relation to mathematical induction. The finite 
ordinals, also, may be conceived as types of series : for example, the 
ordinal number n may be taken to mean " a serial relation of 7i terms ;'''' 
orj in popular language, n terms in a row. This is an ordinal notion, 
distinct from " /ith,'' and logically prior to it*. In this sense, n is the 
name of a class of serial relations. It is this sense, not that expressed 
by "Tith," which is generalized by Cantor so as to apply to infinite 
series. 

291. Let us begin with Cantor's definition of the second class of 
ordinal numbers f. 

" It is now to be shown," he says, " how we are led to the definitions 
of the new numbers, and in what way are obtained the natural sections, 
which I call classes of numbers^ in the absolutely endless series of real 

integers The series (1) of positive real whole numbers 1, S, 3, ... p,. . . 

arises from repeated positing and combination of units which are pre- 
supposed and regarded as equal ; the number v is the expression both for a 
certain finite amount {AnzaliT) of such successive positings, and for the com- 
bination of the units posited into a whole. Thus the formation of finite 

^ Cf. supra Part IV, Chap, xxix, §§ 231, 232. 
t Mannichfaltigkeitslehre, § 11, pp. 32, 33. 



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290, 291] 2'ransfinite Ordinals 313 

real whole numbers rests on the addition of a unit to a number which 
has already been formed ; I call this moment, which, as we shall see 
immediately, also plays an essential part in the formation of the higher 
integers, the first principle of formation. The amount {Anzalil) of 
possible numbers v of the class (1) is infinite, and there is no greatest 
among them. Thus however contradictory it would be to speak of 
a greatest number ,of the class (1), there is yet nothing objectionable 
in imagining a new number, which we will call w, which is to express 
that the whole collection (1) is given by its law in its natural order 
of succession. (In the same way as v expi^esses the combination of a 
certain finite amount of units into a whole.) It is even permissible to 
think of the newly created number co as a limit, towards which the 
numbers v tend, if by this nothing else is understood but that « is 
the first integer which follows all the numbers v^ i.e. is to be called 
greater than each of the numbers v. By allowing further additions 
of units to follow the positing of the number « we obtain, by the help 
of \h^ first principle of formation, the further numbers 

w-f-lj ft) + S, CO -\- p^ ; 

Since here again we come to no greatest number, we imagine a new one, 
which we may call 2g), and which is to be the first after all previous 
numbers v and co + v. 

"The logical function which has given us the two numbers co and ^co 
is evidently different from the first principle of formation ; I call it the 
second principle of formation of real integers, and define it more exactly 
as follows : If we have any determinate succession of defined real integers, 
among which there is no greatest number, by means of this second 
principle of formation a new number is created, which is regarded as 
the limit of those numbers, i.e. is defined as the next number greater 
than all of them." 

The two principles of formation will be made clearer by considering 
that an ordinal number is merely a type or class of series, or rather 
of their generating relations. Thus if we have any series which has 
no last term, every part of such a series which can be defined as all 
the terms up to and including a certain term of the series will have 
a last term. But since the series itself has no last term, it is of a 
different type from any such part or segment of itself. Hence the 
ordinal number representing the series as a whole must be different 
from that representing any such segment of itself, and must be a 
number having no immediate predecessor, since the series has no last 
term. Thus co is simply the name of the class progression^ or of the 
generating relations of series of this class. The second principle of 
formation, in short, is that by which we define a certain type of series 
having no last term. Considering the ordinals preceding any ordinal 
a which is obtained by the second principle as representing segments 



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314 Infinity and Continuity [chap, xxxviii 

of a series represented by a, the ordinal a itself represents the limit 
of such segments ; and as we saw before, the segments always have a 
limit (provided they have no maximum), even when the original series 
has none*. 

In order to define a class among transfinite ordinals (of which, as 
is evident, the succession is infinite). Cantor introduces what he calls 
a principle of limitation (IIemmung'sprincip)'f , According to this 
principle, the second class of ordinals is to consist only of those whose 
predecessors, from 1 upwards, form a series of the first power, i.e. a series 
whose cardinal number is oCq? or one whose terms, in a suitable order, 
have a one-one relation to the finite integers. It is then shown that 
the power, or cardinal number, of the second class of ordinals as a 
whole, is different from a^ (p. 35), and is further the very next cardinal 
number after ao (p- 37). What is meant by the next cardinal number 
to tto results clearly from the following proposition (p. 38) : " If M be any 
well-defined collection of the power of the second class of numbers, 
and if any infinite portion M' of J/ be taken, then either the collection 
M^ can be considered as a simply infinite series, or it is possible to 
establish a unique and reciprocal correspondence between M and M'.'''' 
That is to say, any part of a collection of the second power is either 
finite, or of the first power, or of the second ; and hence there is no 
power between the first and second. 

292. Before proceeding to the addition, multiplication, etc.^ of 
ordinals, it will be well to take the above propositions, as far as possible, 
out of their mathematical dress, and to state, in ordinary language, 
exactly what it is they mean. As for the ordinal &>, this is simply the 
name for the class of generating relations of progressions. We have 
seen how a progression is defined : it is a series which has a first term, 
and a term next after each term, and which obeys mathematical induc- 
tion. By mathematical induction itself we can show that every part 
of a progression, if it has a last term, has some finite ordinal number n, 
where n denotes the class of series consisting of n terms in order ; while 
every part which has no last term is itself a progression ; also we can 
show (what is indeed obvious) that no finite ordinal will represent a 
progression. Now progressions are a perfectly definite class of series, 
and the principle of abstraction shows that there is some entity to 
which all of them have a relation which they have to nothing else — 
for all progressions are ordinally similar (i.e. have a one-one relation 

■^ On the segments of well-ordered series see Cantor's article in Math. Annalen, 
xLix^ § 13. It is important to observe that the ordinals above explained are 
analogous^ in their genesis^ to the real numbers considered as segments {vide Chap, 
xxxiii^ supra). Here^ as there^ the existence of co is not open to question when 
the segment-theory is adopted^ whereas on any other theory the existence-theorem 
is indemonstrable and unplausible. 

f Mannichfaltigkeitslehre, p. 34. 



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291-293] T7^ansfinite Ordinals 315 

such that earher terms are correlated with earlier ones, and later with 
later), and ordinal similarity is symmetrical, transitive, and (among 
series) reflexive. This entity, to which the principle of abstraction 
points, may be taken to be the type or class of serial relations, since 
no series can belong to more than one type of series. The type to 
which progressions belong, then, is what Cantor calls w. Mathematical 
induction, starting from any finite ordinal, can never reach «, since 
CO is not a member of the class of finite ordinals. Indeed, we may define 
the finite ordinals or cardinals — and where series are concerned, this 
seems the best definition— as those which, starting from or 1, can be 
reached by mathematical induction. This principle, therefore, is not to 
be taken as an axiom or postulate, but as the definition of finitude. It 
is to be observed that, in virtue of the principle that every number has 
an immediate successor, we can prove that any assigned number, say, 
10,937, is finite — provided, of course, that the number assigned is a 
finite number. That is to say, every proposition concerning 10,937 
can be proved without the use of mathematical induction, which," as 
most of us can remember, was not mentioned in the Arithmetic of our 
childhood. There is therefore no kind of logical ei;ror in using the 
principle as a definition of the class of finite numbers, nor is there a 
shadow of a reason for supposing that the principle applies to all 
ordinal or all cardinal numbers. 

At this point, a word to the philosophers may be in season. Most 
of them seem to suppose that the distinction between the finite and 
the infinite is one whose meaning is immediately evident, and they 
reason on the subject as though no precise definitions were needed. 
But the fact is, that the distinction of the finite from the infinite is 
by no means easy, and has only been brought to light by modern 
mathematicians. The numbers and 1 are capable of logical definition, 
and it can be shown logically that every number has a successor. We 
can now define finite numbers either by the fact that mathematical 
induction can reach them, starting from or 1— in Dedekind's language, 
that they form the chain of or 1 — or by the fact that they are the 
numbers of collections such that no proper part of them has the same 
number as the whole. These two conditions may be easily shown to be 
equivalent. But they alone precisely distinguish the finite from the 
infinite, and any discussion of infinity which neglects them must be more 
or less frivolous. 

293. With regard to numbers of the second class other than &>, 
we may make the following remark. A collection of two or more 
terms is always, except possibly for some very large infinite collections, 
the field of more than one serial relation. Men may be arranged by 
their rank, age, wealth, or in alphabetical order : all these relations 
among men generate series, and each places mankind in a different 
order. But when a collection is finite, all possible orders give one and 



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316 Infinity and Continuity [chap, xxxyiii 

the same ordinal number, namely that corresponding to the cardinal 
nmiiber of the collection. That is to say, all series which can be formed 
of a certain finite number of terms are ordinally similar. With infinite 
series, this is quite different. An infinite collection of terms which is 
capable of different orders may belong, in its various orders, to quite 
different types. We have already seen that the rationals, in one order, 
form a compact series with no beginning or end, while in another order 
they form a progression. These are series of entirely different types ; 
and the same possibility extends to all infinite series. The ordinal type 
of a series is not changed by the interchange of two consecutive terms, 
nor, consequently, in virtue of mathematical induction, by any finite 
number of such interchanges. The general principle is, that the type 
of a series is not changed by what may be called s^ permutation. That 
is, if P be a serial relation by which the terms of u are ordered, R a 
one-one relation whose domain and whose converse domain are both u^ 

then RPR is a serial relation of the same type as P ; and all serial 
relations whose field is u^ and w^hich are of the same type as P, are of 
the above form RPR. But by a rearrangement not reducible to a 
permutation, the type, in general, is changed. Consider, for example, 
the natural numbers, first in their natural order, and then in the order 
in which S comes first, then all the higher numbers in their natural 
order, and last of all 1. In the first order, the natural numbers form 
a progression ; in the second, they form a progression together with 
a last term. In the second form, mathematical induction no longer 
applies ; there are propositions which hold of 2, and of every subsequent 
finite number, but not of 1. The first form is the type of any funda- 
mental series of the kind we considered in Chapter xxxvi ; the second is 
the type of any such series together with its limit. Cantor has shown 
that every denumerable collection can be given an order which corre- 
sponds to any assigned ordinal number of the second class*. Hence 
the second class of ordinal numbers may be defined as all the types 
of well-ordered series in which any one given denumerable collection can 
be arranged by means of different generating relations. The possibility 
of such different types depends upon the fundamental property of infinite 
collections, that an infinite part of an infinite collection can always 
be found, which will have a one-one correlation with the whole. If 
the original collection was a series, the part, by this correlation, becomes 
a series ordinally similar to the whole : the remaining terms, if added 
after all the terms of the infinite part, will then make the whole 
ordinally different from what it wasf. 

■^ Acta Math, ii, p. 394. 

t The remaining terms_, if they be finite in number^ will often not alter the type 
if added at the beginning ; but if they be infinite^ they will in general alter it even 
then. This will soon be more fully explained. 



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293,294] Transfinite Ordinals 317 

We may assimilate the theory of ordinals to that of cardinals as 
follows. Two relations will be said to be like when there is a one-one 
relation S^ v/hose domain is the field of one of them (P), and which 
is such that the other relation is SFS. If P be a well-ordered relation^ 
i.e. one which generates a well-ordered series, the class of relations like 
P may be defined as the ordinal number of P. Thus ordinal numbers 
result from likeness among relations as cardinals from similarity among 
classes. 

294. We can now understand the rules for the addition and multi- 
plication of transfinite ordinals. Both operations obey the associative, 
but not the commutative law. The distributive law is true, in general, 
only in the form 7 (a + /3) = 7a -h 7/8, 

where a-f-^, a, ^ are multipliers*. That addition does not obey the 
commutative law may be easily seen. Take for example o) + 1 and 
1 -f (w. The first denotes a progression followed by a single term : 
this is the type presented by a progression and its limit, which is 
different from a simple progression. Hence co -h 1 is a different ordinal 
from CO. But 1 + « denotes a progression 'preceded by a single term, 
and this is again a progression. Hence 1 + « = «, but 1 -1- o) does not 
equal « -f 1 f. The numbers of the second class are, in fact, of two 
kinds, (1) those which have an immediate predecessor, (2) those which 
have none. Numbers such as «, « . 2, a).3,...a}^, w^.-co'^... have no 
immediate predecessor. If any of these numbers be added to a finite 
number, the same transfinite number reappears ; but if a finite number 
be added to any of these numbers, we get a new number. The numbers 
with no predecessor represent series which have no end, while those 
which have a predecessor represent series which have an end. It is 
plain that terms added at the beginning of a series with no end leave it 
endless ; but the addition of a terminating series after an endless one pro- 
duces a terminating series, and therefore a new type of order. Thus there 
is nothing mysterious about these rules of addition, which simply express 
the type of series resulting from the combination of two given series. 

Hence it is easy to obtain the rules of subtraction |. If a is less than 
^, the equation .^ a + f = ^ 

has always one, and only one solution in |, which we may represent by 
13— OL. This gives the type of series that must be added after a to 
produce 13. But the equation 

■^ ManniehfaUigkeitslehre, p. 39; a+/3 will be the type of a series consisting* of 
two parts^ namely a part of the type a followed by a part of the type /3 ; ya will be 
the type of a series consisting of a series of the type a of series of the type y. Thus 
a series composed of two progressions is of the type co . 2. 

+ Math. Annalen, xlvi, § 8. 

J MannichfaltigUeitslehre, p. 39. 



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318 Infinity and Continuity [chaf. xxxvill 

will sometimes have no solution, and at other times an infinite number 
of solutions. Thus the equation 

f + Cl> = ft) + 1 

has no solution at all : no number of terms added at the beginning of 
a progression will produce a progression together with a last term. In 
fact, in the equation ^ -|- a = /^^ if a represents an endless type, while ^ 
represents a terminating type, it is sufficiently evident that terms added 
before a can never produce a terminating type, and therefore can never 
produce the type yS. On the other hand, if we consider the equation 

this will be satisfied by f = co + th where n is zero or any finite number. 
For n before the second w will coalesce with this to form w, and thus 
ft) + 71 4- ft) = ft). 2. In this case, therefore, f has an infinite number of 
values. In all such cases, however, the possible values of ^ have a 
minimum, which is a sort of principal value of the difference between ^ 
and a. Thus subtraction is of two kinds, according as we seek a 
number which, added to a, will give /5, or a number to which a may be 
added so as to give /3. In the first case, provided a is less than /3, there 
is always a unique solution ; in the second case, there may be no 
solution, and there may be an infinite number of solutions. 

295. The multiplication of ordinals is defined as follows*. Let 
M and N be two series of the types a and /3. In iV, in place of each 
element n^ substitute a series Mn of the type a ; and let S be the series 
formed of all the terms of all series J/^, taken in the following order : 
(1) any two elements of S which belong to the same series Mn are to 
preserve the order they had in Mn ; two elements which belong to 
different series J/^, Mn' are to have the order which n and n have in N, 
Then the type of S depends only upon a and y^, and is defined to be 
their product ajS^ where a is the multiplicand, and ^ the multiplicator. 
It is easy to see that products do not always obey the commutative law. 
For example, 2 . ft) is the type of series presented by 

which is a progression, so that 2 . g) = g). But ft). 2 is the type 

^1? ^25 ^3 ••• ^l/? •••9 t/l? 4/2? /3? '" Jvi "• 

which is a combination of two progressions, but not a single progression. 
In the former series, there is only one term, ^i, which has no immediate 
predecessor ; in the latter there are two, ei and fi . 

Of division, as of subtraction, two kinds must be distinguished •(•. If 
there are three ordinals a, yS, 7, such that /3 = a^, then the equation 
/3 = a| has no other solution than ^ = 7, and we may therefore denote 

■^ Math, Amialen, xlvi^ § 8. 
+ Mannichfaltigkeitslehre, p. 40. 



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294-296] Transfinite Ordinals 319 

7 by ^/a*. But the equation /3 = fa, if soluble at all, may have several 
or even an infinity of roots ; of which, however, one is always the 
smallest. This smallest root is denoted by iSJIoL. 

Multiplication of ordinals is the process of representing a series of 
series as a single series, each series being taken as a whole, and preserving 
its place in the series of series. Division, on the other hand, is the 
process of splitting up a single series into a series of series, without 
altering the order of its terms. Both these processes have some im- 
portance in connection with dimensions. Division, as is plain, is only 
possible with some types of series ; those with which it is not possible 
may be called primes. The theory of primes is interesting, but it is 
not necessary for us to go into itf. 

296. Every rational integral or. exponential function of co is a 
number of the second class, even when such numbers as w'^, 0)"^ etc.^ 
occur^. But it must not be supposed that all types of denumerable 
series are capable of such a form. For example, the type ?;, which 
represents the rationals in order of magnitude §, is wholly incapable of 
expression in terms of co. Such a type is not called by Cantor an 
ordinal number. The term ordinal numher is reserved for well-ordered 
series, i.e. such as have the following two properties || : 

I. There is in the series F a first term. 

II. If F' is a part of F^ and if F possesses one or more terms which 

come after all the terms of F\ then there is a term f of 
F which immediately follows F\ so that there is no term of 
F before y^ and after all terms of F' . 
All possible functions of co and finite ordinals only, to the exclusion of 
other types such as that of the rationals, represent well-ordered series, 
though the converse does not hold. In every well-ordered series, there 
is a term next after any given term, except the last term if there be 
one ; and provided the series is infinite, it always contains parts which 
are progressions. A term which comes next after a progression has 
no immediate predecessor, and the type of the segment formed of its 
predecessors is of what is called the second species. The other terms 
have immediate predecessors, and the types of the segments formed of 
their predecessors are said to be of the first species. 

■^ Cantor has changed his notation in regard to multiplication : formerly^ in a . /3_, 
a was the multiplicator^ and /3 the multiplicand; now_, the opposite order is 
adopted. In following older works^ except in actual quotations^ I have altered the 
order to that now adopted. 

t See Ma7i7iich/altigkeAtslehre, p. 40. 

X On the exponential function^ see Math. Annalen, xlix^ §§ 18-20. 

§ Math Annaleriy xlvi^ § 9. 

II Main. Annalen, xlix, § 12. The definition may be replaced by the following, 
which is equivalent to it: a series is well-ordered if every class contained in the 
series (except of course the null-class) has a first term. 



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320 Infinity and Continuity [chap, xxxviil 

297. The consideration of series which are not well-ordered is im- 
portant, though the results have far less affinity to Arithmetic than in 
the case of well-ordered series. Thus the type 77 is not expressible as a 
function of cw, since all functions of « represent series with a first term, 
whereas r) has no first term, and all functions of &) represent series in 
which every term has an immediate successor, which again is not the 
case with r). Even the series of negative and positive integers and zero 
cannot be expressed in terms of o), since this series has no beginning. 
Cantor defines for this purpose a serial type ^o), which may be taken as 
the type of a regression (ib. § 7). The definition of a progression, as 

we have seen, is relative to some one-one aliorelative Pf. When P 

generates a progression, this progression with respect to P is a regression 
with respect to P, and its type, considered as generated by P, is denoted 
by *a). Thus the whole series of negative and positive integers is 
of the type ^co + oy. Such a series can be divided anywhere into two 
progressions, generated by converse relations ; but in regard to one 
relation, it is not reducible to any combination of progressions. Such a 
series is completely defined, by the methods of Part IV, as follows : P is 

a one-one aliorelative ; the field of P is identical with that of P ; the 
disjunctive relation "some finite positive power of P^' is transitive and 
asymmetrical ; and the series consists of all terms having this relation 
or its converse to a given term together with the given term. The class 
of series corresponding to any transfinite ordinal type may always be 
thus defined by the methods of Part IV ; but where a type cannot be 
expressed as a function of « or * o) or both, it will usually be necessary, 
if we are to define our type completely, either to bring in a reference to 
some other relation, in regard to which the terms of our series form a pro- 
gression, or to specify the behaviour of our series with respect to limits. 
Thus the type of the series of rationals is not defined by specifying that 
it is compact, and has no beginning or end ; this definition applies also, 
for example, to what Cantor calls the semi-continuum, i.e. the continuum 
with its ends cut off. We must add that the rationals are denumerable, 
i.e. that, with respect to another relation, they form a progression.. 
I doubt whether, in this case, the behaviour of the rationals with 
regard to limits can be used for definition. Their chief characteristics 
in this respect are : (1) that they are condensed in themselves, i.e. every 
term of them is the limit of certain progressions and regressions ; 
(2) in any interval, a progression or a regression which has no limit is 
contained. But both these characteristics belong to the series of 
irrational numbers, i.e. to the series obtained by omitting all rationals 
from the series of real numbers ; yet this series is not denumerable. 

t An aliorelative is a relation which no term can have to itself. This term is due 
to Pierce. See Schroder^ Algebra u. Logik der Relative, p. 181. 



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297-299] 2\ansjmite Ordinals 321 

Thus it would seem that we cannot define the type 77, to which the 
rationals belong, without reference to two generating relations. The 
type 77 is that of endless compact series whose terms, with reference to 
another relation, form a progression. 

From the last remark, we see clearly the importance of the correlation 
of series, with which we began the discussions of Part V. For it is only 
by means of correlation that the type of the rationals, and hence the 
continuum, can be defined. Until we bring in some other relation than 
that by which the order of magnitude among rationals arises, there 
is nothing to distinguish the type of the rationals from that of the 
irrationals. 

298. The consideration of ordinals not expressible as functions of 
w shows clearly that ordinals in general are to be considered — as I 
suggested at the beginning of this chapter — as classes or types of 
serial relations, and to this view Cantor himself now apparently adheres; 
for in the article in the Mathematische Annalen^ Vol. xlvi, he speaks of 
them always as types of order, not as numbers, and in the following 
article {Math. Annalen^ xlix, § IS), he definitely restricts ordinal numbers 
to well-ordered series. In his earlier writings, he confined himself 
more to functions of w, which bear many analogies to more familiar 
kinds of numbers. These are, in fact, types of order which may be 
presented by series of finite and transfinite cardinals which begin with 
some cardinal. But other types of order, as we have now seen, have 
very little resemblance to numbers. 

299. It is worth while to repeat the definitions of general notions 
involved in terms of what may be called relation -arithmetic*. If P, Q be 
two relations such that there is a one-one relation S whose domain is the 

field of P and which is such that Q = SPS^ then P and Q are said to be 
like. The class of relations like P, which I denote by XP, is called 
P's relation-numher. If the fields of P and Q have no common terms, 
P + Q is defined to be P or Q or the relation which holds between any 
term of the field of P and any term of the field of Q, and between no 
other terms. Thus P 4- Q is not equal to Q + P. Again \P -h XQ is 
defined as X (P -|- Q). For the summation of an infinite number of 
relations, we require an aliorelative whose field is composed of relations 
whose fields are mutually exclusive. Let P be such a relation, p its 
field, so that ^ is a class of relations. Then 2pp is to denote either one 
of the relations of the class p or the relation of any term belonging 
to the field of some relation Q of the class ^ to a term belonging to the 
field of another relation R (of the class p) to which Q has the relation P. 
(If P be a serial relation, and p a class of serial relations, Spp will be the 
generating relation of the sum of the various series generated by terms 
of p taken in the order generated by P.) We may define the sum 

^ Cf. Part IV, Chap, xxix, § 231. 
R, 21 



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322 Infinity and Continuity [chap, xxxyiii 

of the relation-numbers of the various terms of p as the relation-number 
of Spp. If all the terms oi p have the same relation -number, say a, 
and if /3 be the relation-number of P, a x /3 will be defined to be the 
relation-number of 2pp. Proceeding in this way, it is easy to prove 
generally the three formal laws which hold of well-ordered series, 
namely : 

(a + ^) + 7 = a + (/3 + 7) 

« (i^ + 7) = '^/^ + ^7 
(a^)7 = a(^7). 

The proofs are very closely analogous to those discovered by Mr White- 
head for cardinal numbers {Amer. Journal of Math.^ Vol. xxiv) ; but 
they differ by the fact that no method has yet been discovered for 
defining an infinite product of relation-numbers, or even of ordinal 
numbers. 

300. It is to be observed that the merit of the above method is 
that it allows no doubt as to existence-theorems — a point in which 
Cantor's work leaves something to be desired. As this is an important 
matter, and one in which philosophers are apt to be sceptical, I shall 
here repeat the argument in outline. It may be shown, to begin with, 
that no finite class embraces all terms : this results, with a little care, 
from the fact that, since is a cardinal number, the number of numbers 
up to and including a finite number n is n-\-l. Further, if n be a 
finite number, n 4- 1 is a new finite number different from all its prede- 
cessors. Hence finite cardinals form a progression, and therefore the 
ordinal number co and the cardinal number ocq exist (in the mathematical 
sense). Hence, by mere rearrangements of the series of finite cardinals, 
we obtain all ordinals of Cantor's second class. We may now define the 
ordinal number cd^ as the class of serial relations such that, if u be a class 
contained in the field of one of them, to say that u has successors implies 
and is implied by saying that it has «« terms or a finite number of terms ; 
and it is easy to show that the series of ordinals of the first and second 
classes in order of magnitude is of this type. Hence the existence of coi 
is proved; and a^ is defined to be the number of terms in a series 
whose generating relation is of the type co^. Hence we can advance to 
0)2 and a^ and so on, and even to o)^ and otc^, whose existence can be 
similarly proved : w^ will be the type of generating relation of a series 
such that, if 11 be a class contained in the series, to say that u has 
successors is equivalent to saying that u is finite or has, for a suitable 
finite value of ?z, a^ terms. This process gives us a one-one correlation 
of ordinals with cardinals : it is evident that, by extending the process, 
we can make each cardinal which can belong to a well-ordered series 
correspond to one and only one ordinal. Cantor assumes as an axiom that 
every class is the field of some well-ordered series, and deduces that all 
cardinals can be correlated with ordinals by the above method. This 



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299-302] Transfinite Ordinals 323 

assumption seems to me unwarranted, especially in view of the fact that 
no one has yet succeeded in arranging a class of 2«o terms in a well- 
ordered series. We do not know that of any two different cardinal 
numbers one must be the greater, and it may be that S'*o is neither 
greater nor less than a-^ and ag and their successors, which may be called 
well-ordered cardinals because they apply to well-ordered classes. 

301. There is a difficulty as regards the type of the whole series of 
ordinal numbers. It is easy to prove that every segment of this series 
is well-ordered, and it is natural to suppose that the whole series is also 
well-ordered. If so, its type would have to be the greatest of all ordinal 
numbers, for the ordinals less than a given ordinal form, in order of 
magnitude, a series whose type is the given ordinal. But there cannot be 
a greatest ordinal number, because every ordinal is increased by the 
addition of 1. From this contradiction, M. Burali-Forti, who dis- 
covered it*, infers that of two different ordinals, as of two different 
cardinals, it is not necessary that one should be greater and the 
other less. In this, however, he consciously contradicts a theorem of 
Cantor's which affirms the opposite f. I have examined this theorem 
with all possible care, and have failed to find any f